\[\begin{matrix} S_k & T_k & \mathrm{OEIS}\\ \\ 2k-1 & 1,-3,24,-297,4896,-100278 & \approx A213641*3*k\\ k^2 & 1,4,52,1252,47380,2589892\\ k^3 & 1,8,280,23624,3923416,1136642696\\ P[k+1]-P[k] & 1,-2,8,-48,352,-2880,25216,-231168,2194944(?),-21492736(?),216926208... & A054726 (?)\\ 1,2,2,4,2,4,2,4,... & 1,-2,8,-48,352,-2880,25216,-231168,2190848,-21292032,211044352 & A054726 \\ P[k+2]-P[k] & 3,12,120,1632,25824,446592,8184576,156605952\cdots 1,4,40,544,8608 & 2^k\cdot Seq[k]?\\ \frac{T(P[k+2]-P[k])}{2^k} & \frac{3}{2},3,15,102,807,6978,63942 P[k+2]-P[k+1]+P[k] & 4,20,280,5540,139120,4238360 \cdots 1,5,70,1385,34780,1059590\\ P[k]-k & 1,1,3,15,117,1341,20943\\ BernoulliB[k] & -1/2, 1/12, -1/72, 1/432, -1/2592, 1/15552, -1/93312 & Sign/Rcip A167747\\ \phi(6^k) & 2,-24,2016,-915840\\ \phi(k+7)-\lambda(k+7) & 2, BernoulliB[2k] & 1/6,1/180,1/18900,-31/7938000,989/9168390000\\ 1/BernoulliB[2k] & 6,180,-2160,-200880\\ 1/(BernoulliB[k]+1) & 2,-12/7,156/49,-76452/9947\\ k! & 1,2,16,416,47104,\cdots\\ \sqrt{k} & 1,\sqrt(2),2+\sqrt{6},-5\sqrt{2}-4\sqrt{3}-2\sqrt{6},22+12\sqrt{2}+8\sqrt{3}+13\sqrt{6}+2\sqrt{30} \\ \psi(k) & \gamma,(1-\gamma)\gamma, \frac{-1}{2}\gamma(5-9\gamma+4\gamma^2)\\ \log(k+1) & \ln(2),\ln(2)\ln(3),\ln(2)\ln(3)\ln(12), \\ 2k & 2,-8,80,-1184,22592 \\ k^2+k-1 & 1,-5,80,-2325,104080 \\ e^k & \frac{1}{e},\frac{-1}{e^3},\frac{1+e}{e^6},-\frac{1+e+2e^2+e^3}{e^{10}},\frac{1+e+2e^2+3e^3+3e^4+3e^5+e^6}{e^{15}}\\ 1/k & 1,-1/2,5/12,-7/18,1631/4320,-96547/259200, 40291823/108864000 & A176599/???\\ k^k & 1,-4,124,-31492,95311228\\ k^(1/k) & 1,-\sqrt{2},2+\sqrt{2}3^(1/3),-2\sqrt{2}-6*3^(1/3)-\sqrt{2}*3^(2/3)\\ \zeta(k+1) & \zeta(2),-\zeta(2)\zeta(3),\zeta(2)\zeta(3)(\zeta(4) + \zeta(3)),-\zeta(2)\zeta(3)(\frac{7}{6}\zeta(8) + \zeta(3)^2 + \zeta(4)(2\zeta(2)+\zeta(5))) \\ f(k)& f(1), \\&-f(1)f(2), \\&f(1)f(2)(f(2)+f(3)), \\&-f[1]f[2](f[2]^2 + 2f[2]f[3] + f[3](f[3]+f[4])), \\& f(1)f(2)( f(2)^3 + 3f(2)^2f(3) + f(2)f(3)(3f(3)+2f(4))+f(3)(f(3)^2 + 2f(3)f(4) + f(4)(f(4)+f(5))))\\ \phi(k)&1,-1,3,-13,75,-525,4219,-38141 & A007178??\\ \lambda(k)&1,-1,3,-13,75,-525,4219,-37757 & A007178??\\ \lambda*(k)& 1,-1,3,-13,75,-525,4347,\\ 1,1,2,2,3,3... & 1,-1,3,-13,71,-461,3447,-29092, & A158882,A003319,A167894\\ 1,2,1,2,1,2,1... & 1,-2,6,-22,90,-394,1806,-8558, & A006318 (Schroder numbers)\\ \beta(k,k+1) & \frac{1}{2},-\frac{1}{24},\frac{1}{240},\frac{-169}{403200},\frac{42851}{1016064000}\\ \frac{\Gamma(k+1)}{\Gamma(k+2)}& \frac{1}{2}, \frac{-1}{6},\frac{7}{72}, -\frac{281}{4320}, \\ \frac{\Gamma(k+2)}{\Gamma(k+1)}& 2,-6,42,-414,5058,-72486,1182762,-21573054 & A115974 (more QED)\\ \frac{\Gamma(k+3)}{\Gamma(k+1)}& 6,-72,2304,-116928 \\ \frac{\Gamma(k+3)}{\Gamma(k+2)}& 3,-12,108,-1332,19908,-342252,6583788 & A127059 \\ \frac{\Gamma(k+3/2)}{\Gamma(k+1/2)}& 3/2,-15/4,45/2,-3105/16,16875/8, \\ \frac{\frac{4}{3\sqrt{\pi}}\Gamma(k+3/2)}{\Gamma(k+1)}& 1, -5/4, 325/96, -224125/18432 \\ Catalan(k) & 1,2,14,238,10486,1360142 & ??Featur in A092269\\ 1,2,3,1,2,3,1,2,3... & 1,-2,10,-56,328,-1988,12400,...\\ 1,2,3,2,3,2,3,2,3... & 1,-2,10,-62,430,-3194,24850,... & A107841\\ 1,2,3,11/3,140/33,...& 1,2,10,72,644 & A177384 \\ (k+1)/k & 2,3,17/2,-349/12,7835/72,-115699/270\\ 1,-1,1,-1,1,-1 & 1,1,0,-1,0,2,0,-5,0,14,... & A090192\\ & 0,\\ 1,2,1,3,1,4,1,5 & 1,-2,6,-24,114,-606,3504, & A134664??\\ 1,2,1,3,1,4,7/12,~164/25... & 1,-2,6,-24,114,-606,3494, & A134664\\ 1,2,2,3,3,4,4,5,5,.. & 1,-2,8,-44,296,-2312,20384 & A111537\\ 1,1,2,1,1,3,1,1,4,.. & 1,-1,3,-11,43,-177 \\ 1,-1,2,-2,3,-3,.. & 1,1,-1,-3,11,29,-221,-531\\ 1,-2,3,-4,5,-6,.. & 1,2,-2,-22,70,866\\ 3,1,1,1,1,1.. & 3,-3,6,-15... & 3xCarmich \\ 1,3,1,1,1,1.. & 1,-3,12,-51,222,-978,4338... & A007854\\ 1,2,0,0,0,0...& 1,-2,4,-8,16.. & A000079 \\ 1,3,0,0,0,0...& 1,-3,9,-27,.. & A000244 \\ 1,3,\frac{2}{3},\frac{29}{6},\frac{51}{58},\frac{4656}{493},\frac{-8120}{4947},\frac{149923}{16296}& 1,-3,11,-50,274,-1764,13068,-109584.. & A000254 by (2,29,51,4656,-8120,149923)/(1,1,3,6,58,493,4947,16296) \\ \\ 2,\frac{3}{2},\frac{1}{6},\frac{-8}{3},\frac{3}{2},\frac{3}{2},0,FAIL & 2,-3,5,-7,11,-13,-7,-13,107,-295\\ 1,2,-1/2,-1/2,-2,2,1,0,0,0,0,0 & 1,-2,3,-5,7,-11,13,-17,7,13,-107,295,-833,1909,... & Unrepresentable \\ 4,\frac{9}{4},\frac{19}{36},\frac{-736}{171},\frac{990}{437},\frac{4731}{2530},\frac{2139}{9130}..., squares of primes.\\ 1,31,\frac{-840}{31},\frac{-7}{248},\frac{-2945}{56},\frac{6448}{133},\frac{-277727}{1607970},\frac{-50887164599}{23504036010} & 1,-31,121,-496,781,-3751,2801,-7936,\cdots & A160893 \\ \end{matrix}\]
\[\begin{matrix} 1,1,1,2,2,3,3,4,4,5,5,... & 1,1,2,6,24,120,720,5040,40320, & A000142 \\ 1,2,1,3,2,4,3,5,4,6,5,... & 1,2,6,24,120,720,5040,40320, & A000142 \\ 1,1,2,3,4,5,6,... & 1,-1,3,-15,105,-945,10395 & A001147\\ 1,1,3,4,6,7,9,10,12,...& ... &A007559 \\ 1,2,1,6,1,55& 1,-2,6,-30,210,-2310 & ..\\ 1,1,1,-1,0,0,0,0,0,0 & Fibbonacci & A000045\\ 1,2,1/2,-1/2,0,0,0,0 & 1, 2, 5, 12, 29, 70, 169, 408 & A000129 \\ 4,15,-1,14/15,-1/210,1/14,0,0,0 & convergents of sqrt(3) & A123480 \\ 1,4,1/4,-1/4,0,0,0,0 & 1,4,17,72,305,1292 & A001076 \\ 1,5,1/5,-1/5,0,0,0,0 & & A052918\\ 1,1,2,-2,0,0,0 & Jacobsthal Numbers & A001045\\ 1,4,\frac{-7}{4},\frac{17}{28},\frac{-32}{119},\frac{7}{17},0,0,0,0,0,0 & 1,4,9,16,... & A000290 \\ 1,1,1,0,0,0,0 & 1,1,2,4,8,16,32 & A011782 \\ 1,5,-6/5,6/5,0,0,0& 3^n-2^n & A001047\\ 2,5/2,1/10,12/5,0,0& 2^n+3^n & A007689\\ 1,2/2,0,0,0 & 1^n & A000012\\ 2,3/2,1/6,4/3,0,0,0 & 1^n + 2^n & A000051\\ 3,4/2,2/6,5/3,1/5,18/10,0,0& 1+2^n+3^n & A001550\\ 4,5/2,3/6,6/3,2/5,21/10,3/14,16/7,0,0& 1+2^n+3^n+4^n & A001551\\ 5,6/2,4/6,7/3,3/5,24/10,6/14,18/7,2/9,25/9,0,0& 1^n \cdots 5^n & A001552\\ \end{matrix}\]
N.B. Can’t seem to represent the Tribonacci sequence. Representing sequences \(1^n .. x^n\), seem to follow a pattern \[2(x+0)/2,(x+1)/2,(x-1)/6,2(x+2)/6,2(x-2)/10,3(x+3)/10,3(x-3)/14,4(x+4)/14,4(x-4)/18,5(x+5)/18...\] which means the sequence for any expression of the form \[\underset{k=1}{\overset{2K_0}{\large K \normalsize}} \frac{g(k,K_0)}{x} =\sum_{n=0}^\infty \frac{(-1)^n}{x^{2n+1}}\sum_{k=1}^{K_0} k^n\\ g(k,x)=\large \Bigg\{ \normalsize \begin{matrix} x, & k=1 \\ \frac{k(2x+k)}{8(k-1)}, & k=\mathrm{even} \\ \frac{(k-1)(2x-k+1)}{8k}, & \mathrm{otherwise} \end{matrix}\]
We see that \(\phi(k)\) and \(\lambda(k)\) give the same series which approaches OEIS A007178, until term \(4219\). By using \(\lambda^*=1,1,2,2,4,2,10,\frac{-31}{10}\cdots\), we can achieve A007178. However, this series seems insignificant. The transform may then be close to a desired transform but not perfect, especially for higher terms. w \(x^2\) instead 1,2,4,6,8
\[\begin{matrix} 1,5,-1,2,-2,-1/2,-1/2,0 & A154638 \\ 1,1,2,3/2,7/2,16/7,33/7,35/11,64/11,66/16,110/16,112/22,174/22 & A001710 \\ 1,3,3,5,5,7,7... & A005412 (QED)\\ 1,3,4,5,6,7,8... & A167872 (QED)\\ 1,2,1,12,-3,-10,23 & A115974 (QED)\\ 1,1,6,4,6 & 1,1,3,7,(73) & \approx A005413 (one missing?)\\ 1,1,3,3,5,5,7,7 & A005411 (QED)\\ 1,4,16,61/2?, ... & A214298 (QED)\\ 1,1,1,1,1,8,27/8?,-3131/72 & A224978 (QED)\\ \\ 1,11,-10/11,10/11,0 & A000042\\ \\ 1,0,0,0,0 & A000007\\ 2,1,0,0,0 & A007395\\ 3,2,1,0,0 & A110593\\ 4,3,2,1,0 & 4\times A084120\\ 5,3,1,0,0 & 5\times A002001\\ 1,2,-3/2,-1/2,0 & A000034\\ 1,2,-1/2,7/2,-36/7,1/7,0 & A010882\\ 1,2,-1/2,-1/2,1,0 & A008619\\ 1,1,1,-2,0 & A016116 \\ 1,6,-4/3,4/3,0 & A171476 \\ 1,2,-1/2,1/2,0 & A000027 \\ 1,1,1,1,0 & A001519\\ 1,1,1,1,1,0 & A124302 \\ 1,1,1,1,1,1,0 & A080937 \\ 1,1,1,1,1,1,1,0 & A024175 \\ 1,1,1,1,1,1,1,1,0 & A080938 \\ 1,1,1,1,1,1,1,1,1,0 & A033191 \\ 1,1,1,1,1,1,1,1,1,1,0 & A211216 \\ 1,1,3,-3,0 & A006130 \\ 1,1,4,-4,0 & A006131 \\ 1,3,1,3,1,3,1,3... & A047891\\ 1,4,1,4,1,4,1,4... & A082298\\ 1,1,2,1,2,1,2,1... & A001003\\ 1,2,2,2,2,2,2,2... & A151374\\ 1,5,2/5,13/5,10/13,29/13,26/29,61/29,58/61,125/61... & A134425\\ 1,1/2,1/2,0& 1,1/2,1/2,1/2,1/2...\\ 1,1/2,1/3,0& 1,1/2,5/12,25/72,125/432,625/2592 & A000351/A167747 \\ 1,1/a,1/b,0& (a+b)^n/\phi((ab)^n) ?? \\ 1,1/2,1/5,0& 7^n/(2\cdot 10^n)\\ 1,1/a,1/b,0& (a+b)^n/((2*a*b)^) ?? \\ 1,2,-1/2,5/2,-16/2,1/5,0 & A068073 \\ 1,-1,1,-1,-1 & \mathrm{q-Catalan}\; (q=-1)\\ 1,1,2,4,8,16,... & A015083 \\ 1,1,3,9,27,81,...& A015084 \\ 1,1,n,n^2,n^3,...& \mathrm{q-Catalan}\; (q=n)\\ 1,1,2,2,4,3,6,4,8,5,... & \mathrm{Fubini}\;\;A000670\\ 1,1,3,2,6,3,9,4,12,5,... & A122704\\ 1,1,1,2,1,3,1,4,1,5,... & A074664 \\ 1,1,1,1,2,1,3,1,4,1,5... & \mathrm{Bell} A000110\\ 1,2,1,2,2,2,3,2,4,2,5... & A001861\\ 1,3,1,3,2,3,3,3,4,3,5... & A027710\\ 1,4,1,4,2,4,3,4,4,4,5... & A078944\\ 1,2,2,0 & A081294 \\ 1,2,2,2,0 & A154626 \\ 1,2,2,2,2,0 & A092807 \\ 1,4,-5/2,1/10,2/5,0 & A103517\\ 1,3,-1,2/3,-1/6,1/2,0 & \mathrm{Triangle} A000217 \\ 1,4,-3/2,5/6,-1/3,3/5,-1/10,1/2,0 &\mathrm{Tetrahed.} A000292\\ 1,2D/2,-D/2,.../6,-.../6,.../10,-.../10, & \mathrm{Gen^{zd} Tri}\\ \end{matrix}\]
There is potentially a very powerful concept available here, where we will be able to write a functional form of sorts for expressions such as \(n \;\mod\;m\).
It is becoming increasingly clear that there may be two sequences involved in describing certain hard to fit members. Then the input series is actually of the form \[C_0,a_1,b_1,a_2,b_2,a_3,b_3...\] where \(C_0\) is just a leading coefficient, and then the two sequences \(a\) and \(b\) progress with their description. Then many sequences will start with \(C_0=1\), if they start with \(1\). Then sequence \(A134425\) actually has two input sequences \[a=5,13/5,29/13,61/29,125/61...\\ b=2/5,10/13,26/29,58/61...\\\] We then see that for the sequence \[t_i=1,5,13,29,61,125... = 2^{i+1}-3,\;i=1,2,3,4\] we can define \[a_i = \frac{t_{i+1}}{t_{i}}\] it then follows that \[b_{i}=\frac{2}{a_i}\] we then have a fully predictable input sequence to generate the output \(A134425\).
Sequence A261518 appears to be created by the input sequence \[1,1,1,2,1,3,44/3,-1567/132,-124365/68948, 1917216532/64959985, -17257891916243/1806322984865...\] This appears to grow randomly, however upon close inspection we may notice if we write the sequence as \[\frac{1}{1},\frac{1}{1},\frac{1}{1},\frac{2}{1},\frac{2}{2},\frac{12}{4},\frac{352}{24},-\frac{50144}{4224},-\frac{31837440}{17650688}, \frac{47117513490432}{1596456591360}, -\frac{14332199328981404614656}{1500101008700819374080}...\] we note that the product of any last two terms’ numerators is the denominator of the next.
We find that the Binomial transform of Fine’s sequence A033321, is generated by the split reciprocal transform of sequence A001519 given by \[1, 1, 2, 5, 13, 34, 89,...\] and the split-reciprocal transform results in the series \[1,1,1,2,1/2,5/2,2/5,13/5,5/13,34/13,13/34,...\]
We find that this reciprocal transform series on the sequence \(1,2,3,4,5,6,7...\) gives another output of the Catalan numbers, without the initial \(1\).
We see that an input sequence of the form \[1,n,1,n,2,n,3,n,4,n,5...\] goes on to have coefficients that are the Bell numbers of order \(n\). These Bell numbers are given by \[B_n=e^{-1}_{n-1}F_{n-1}\left(2,...,2;1,...,1;1\right)\] where the \(2\) and \(1\) are repeated \(n-1\) times [http://mathworld.wolfram.com/BellNumber.html]. Then we may write \[\mathcal{I}(1;n;1,2,3,4,\cdots)=B_n\]