A290315 - OEIS

A290315

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 A154537 (S2[2,1] generalized Stirling2), for n >= 0.

1, 1, 2, 1, 16, 12, 1, 66, 284, 120, 1, 224, 2872, 5952, 1680, 1, 706, 21080, 116336, 146064, 30240, 1, 2160, 132228, 1531072, 4804656, 4130304, 665280, 1, 6530, 760500, 16271080, 101422640, 208791648, 132557760, 17297280, 1, 19648, 4155120, 151922560, 1661273440, 6556459008, 9657333504, 4766423040, 518918400, 1, 59010, 21993776, 1304454880, 23155279200, 155184721088, 427142449920, 477104352768, 189945688320, 17643225600

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OFFSET

0,3

COMMENTS

The ordinary generating function (o.g.f.) of the (n+1)-th diagonal sequence of the Sheffer triangle A154537 = (e^x, e^(2*x) - 1), called S2[2,1], is GS2(2,1;n,x) = P(n, x)/(1 - 2*x)^(2*n+1), with the row polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, n >= 0.

In the general case of 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) the o.g.f. of the (n+1)-th diagonal sequence is G(d,a;n,x) = P(d,a;n,x)/(1 - d*x)^(2*n + 1) with the numerator polynomial P and coefficient table T(d,a;n,k).

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 in A290311.

LINKS

Table of n, a(n) for n=0..54.

Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], 2017.

FORMULA

T(n, k) = [x^k] P(n, x) with the numerator polynomial in the o.g.f. of the (n+1)-th diagonal sequence of the triangle A154537. See a comment above.

EXAMPLE

The triangle T(n, k) begins:

n\k 0 1 2 3 4 5 6 7 ...

0: 1

1: 1 2

2: 1 16 12

3: 1 66 284 120