A290595 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 A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.

1, 1, 2, 4, 19, 4, 28, 222, 147, 8, 280, 3194, 4128, 887, 16, 3640, 55024, 113566, 52538, 4835, 32, 58240, 1107336, 3268788, 2562676, 555684, 25167, 64, 1106560, 25526192, 100544412, 117517960, 45415640, 5301150, 128203, 128, 24344320, 663605680, 3325767376, 5352311764, 3189383200, 695714590, 47537320, 646519, 256, 608608000, 19213911360, 118361719296, 248493947496, 208996478388, 72479948400, 9696965250, 410038434, 3245139, 512

Offset: 0

Author

Wolfdieter Lang, Aug 08 2017

Keywords

• nonn
• tabl

Comments

The ordinary generating function (o.g.f.) of the (n+1)-th diagonal sequence of the Sheffer triangle A286718 = ((1 - 3*x)^(-1/3), -log(1 - 3*x)/3), called |S1hat[3,1]|, is GD(3,1;n,x) = P(n, x)/(1 - x)^(2*n+1), with the row polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, n >= 0.

For the two parameter Sheffer case |S1hat[d,a]| = ((1 - d*x)^{-a/d}, -log(1 - d*x)/d) (with gcd(d,a) = 1, d >=0, a >= 0, and for d = 1 one takes a = 0) the e.g.f. ED(t, x) of the o.g.f.s {GD(d,a;n,x)}_{n>=0} of the diagonal sequences with elements D(d,a;n,m) = |S1hat[d,a]|(n+m, m) (n=0 for the main diagonal) is of interest. It can be computed via Lagrange's theorem. For the special Sheffer case (1, f(x)) this has been done by P. Bala (see the link). This method can be generalized for Sheffer (g(x), f(x)), as shown in the W. Lang link.

Examples

			The triangle T(n, k) begins:
n\k        0        1         2         3        4       5      6   7 ...
0:         1
1:         1        2
2:         4       19         4
3:        28      222       147         8
4:       280     3194      4128       887       16
5:      3640    55024    113566     52538     4835      32
6:     58240  1107336   3268788   2562676   555684   25167      6
7:   1106560 25526192 100544412 117517960 45415640 5301150 128203 128
...
n = 8: 24344320 663605680 3325767376 5352311764 3189383200 695714590 47537320 646519 256,
n = 9: 608608000 19213911360 118361719296 248493947496 208996478388 72479948400 9696965250 410038434 3245139 512.
n = 3: The o.g.f. of the 4th diagonal sequence of A286718, [28, 418, 2485, ...] = A024213(n+1), n >= 0, is P(3, x) = (28 + 222*x + 147*x^2 + 8*x^3)/(1 - 3*x)^7.
		

Links

• Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
• Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Crossrefs

Cf. A024213, A286718, A288875 ([2,1] case).

Formula

T(n, k) = [x^k] P(n, x) with the numerator polynomials of the o.g.f. GD(n, x) = P(n, x)/(1-x)^(2*n+1) of the (n+1)-th diagonal sequence of the triangle A286718. See a comment above.