cp's OEIS Frontend

A290316 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.

%I A290316 #13 Jan 25 2020 18:20:26
%S A290316 1,1,6,1,48,90,1,234,2214,2160,1,996,27432,114588,71280,1,4062,260748,
%T A290316 2791800,6770628,2993760,1,16344,2178630,48256344,280652364,454137840,
%U A290316 152681760,1,65490,16966530,691711920,7846782660,29157089832,34236464400,9160905600,1,262092,126820980,8851303620,174637926180,1219804572672,3187159638984,2871984146400,632102486400,1,1048518,924701832,105253405560,3359003385600,39425596747272,188635513271256,369150976563264,265665182896800,49303993939200
%N A290316 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.
%C A290316 The ordinary generating function (o.g.f.) of the (n+1)-th diagonal sequence of the Sheffer triangle A282629 = (e^x, e^(3*x) - 1), called S2[3,1], is  GS2(3,1;n,x) = P(n, x)/(1 - 3*x)^(2*n+1), with the row polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, n >= 0.
%C A290316 For the general case Sheffer S2[d,a] = (e^(a*x), e^(d*x) - 1) (with gcd(d,a) = 1, d >=0, a >= 0, and for d = 1 one takes a = 0) see a comment in A290315.
%C A290316 For the computation of the exponential generating function (e.g.f.) of the o.g.f.s of the diagonal sequences of a Sheffer triangle (lower triangular matrix) via Lagrange's theorem see a comment and link in A290311.
%H A290316 Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], 2017.
%F A290316 T(n, k) = [x^k] P(n, x) with the numerator polynomials of the o.g.f. of the (n+1)-th diagonal sequence of the triangle A282629. See a comment above.
%e A290316 The triangle T(n, k) begins:
%e A290316 n\k 0     1        2         3          4           5           6          7 ...
%e A290316 0:  1
%e A290316 1:  1     6
%e A290316 2:  1    48       90
%e A290316 3:  1   234     2214      2160
%e A290316 4:  1   996    27432    114588      71280
%e A290316 5:  1  4062   260748   2791800    6770628     2993760
%e A290316 6:  1 16344  2178630  48256344  280652364   454137840   152681760
%e A290316 7:  1 65490 16966530 691711920 7846782660 29157089832 34236464400 9160905600
%e A290316 ...
%e A290316 n = 8: 1 262092 126820980 8851303620 174637926180 1219804572672 3187159638984 2871984146400 632102486400,
%e A290316 n = 9: 1 1048518 924701832 105253405560 3359003385600 39425596747272 188635513271256 369150976563264 265665182896800 49303993939200.
%e A290316 ...
%e A290316 n = 3: The o.g.f. of the 4th diagonal sequence of A282629, [1, 255, 7380, ...], is P(3, x) = (1 + 234*x + 2214*x^2 + 2160*x^3)/(1 - 3*x)^7.
%Y A290316 Cf. A282629, A290311, A290315.
%K A290316 nonn,tabl
%O A290316 0,3
%A A290316 _Wolfdieter Lang_, Aug 08 2017