This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A156308 #47 Aug 08 2025 02:47:31
%S A156308 1,4,1,9,6,1,16,20,8,1,25,50,35,10,1,36,105,112,54,12,1,49,196,294,
%T A156308 210,77,14,1,64,336,672,660,352,104,16,1,81,540,1386,1782,1287,546,
%U A156308 135,18,1,100,825,2640,4290,4004,2275,800,170,20,1
%N A156308 Inverse of triangle S(n,m) defined by sequence A156290, n >= 1, 1 <= m <= n.
%C A156308 From _Wolfdieter Lang_, Jun 26 2011: (Start)
%C A156308 This triangle S(n,m) appears as U_m(n) in the Knuth reference on p. 285. It is related to the Riordan triangle T_m(n) = A111125(n,m) by S(n,m) = A111125(n,m) - A111125(n-1,m), n >= m >= 1 (identity on p. 286).
%C A156308 Also, S(n,m)-S(n-1,m) = A111125(n-1,m-1), n >= 2, m >= 1 (identity on p. 286). (End)
%C A156308 These polynomials may be expressed in terms of the Faber polynomials of A263916 and are embedded in A127677 and A208513. - _Tom Copeland_, Nov 06 2015
%H A156308 G. C. Greubel, <a href="/A156308/b156308.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A156308 Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H A156308 Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2502.13673">Pseudo-involutions in the Riordan group and Chebyshev polynomials</a>, arXiv:2502.13673 [math.CO], 2025.
%H A156308 Johann Cigler, <a href="https://arxiv.org/abs/2508.04751">A bivariate version of the spread polynomials</a>, arXiv:2508.04751 [math.CO], 2025.
%H A156308 D. E. Knuth, <a href="http://arxiv.org/abs/math/9207222">Johann Faulhaber and sums of powers</a>, arXiv:math/9207222 [math.CA], 1992; Math. Comp. 61 (1993), no. 203, 277-294.
%F A156308 S(n, m) = (n/m) * binomial(n + m - 1, 2*m - 1).
%F A156308 From _Peter Bala_, May 01 2014: (Start)
%F A156308 The n-th row o.g.f. is polynomial R(n,x) = 2/x*( T(n,(x + 2)/2) - 1 ), where T(n,x) is Chebyshev polynomial of the first kind. They form a divisibility sequence: if n divides m then R(n,x) divides R(m,x) in the ring Z[x].
%F A156308 R(2*n,x) = (x + 4)*U(n-1,(x + 2)/2)^2;
%F A156308 R(2*n + 1,x) = ( U(n,(x + 2)/2) + U(n-1,(x + 2)/2) )^2.
%F A156308 O.g.f.: Sum_{n >= 0} R(n,x)*z^n = z*(1 + z)/( (1 - z)*(1 - (x + 2)*z + z^2) ). (End)
%F A156308 The polynomial R(n,x) defined above satisfies (x + 1/x - 2) * R(n, x + 1/x - 2) = x^n + 1/x^n - 2. - _Alexander Burstein_, May 23 2021
%e A156308 Triangle starts:
%e A156308 n=1: 1;
%e A156308 n=2: 4, 1;
%e A156308 n=3: 9, 6, 1;
%e A156308 n=4: 16, 20, 8, 1;
%e A156308 ...
%t A156308 S[m_] := Flatten[Table[k/j Binomial[k + j - 1, 2 j - 1], {k, 1, m}, {j, 1, k}]]
%o A156308 (Sage) flatten([[(n/k)*binomial(n+k-1, 2*k-1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 01 2022
%o A156308 (Magma) [(n/k)*Binomial(n+k-1, 2*k-1): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 01 2022
%Y A156308 Same as triangle A208513 with the first column truncated.
%Y A156308 Columns: A000290 (m=1), A002415 (m=2), A040977 (m=3), A053347 (m=4), A054334 (m=5).
%Y A156308 Cf. A078812, A111125, A127677, A208513, A263916.
%K A156308 easy,nonn,tabl
%O A156308 1,2
%A A156308 _Hartmut F. W. Hoft_, Feb 07 2009
%E A156308 Edited by _Max Alekseyev_, Mar 05 2018