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.

1, 1, 6, 1, 48, 90, 1, 234, 2214, 2160, 1, 996, 27432, 114588, 71280, 1, 4062, 260748, 2791800, 6770628, 2993760, 1, 16344, 2178630, 48256344, 280652364, 454137840, 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

Offset

0,3

Comments

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.

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.

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.

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 polynomials of the o.g.f. of the (n+1)-th diagonal sequence of the triangle A282629. 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     6
2:  1    48       90
3:  1   234     2214      2160
4:  1   996    27432    114588      71280
5:  1  4062   260748   2791800    6770628     2993760
6:  1 16344  2178630  48256344  280652364   454137840   152681760
7:  1 65490 16966530 691711920 7846782660 29157089832 34236464400 9160905600
...
n = 8: 1 262092 126820980 8851303620 174637926180 1219804572672 3187159638984 2871984146400 632102486400,
n = 9: 1 1048518 924701832 105253405560 3359003385600 39425596747272 188635513271256 369150976563264 265665182896800 49303993939200.
...
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.

Crossrefs

Cf. A282629, A290311, A290315.

Sequence in context: A136235 A113392 A113387 * A090435 A136237 A308281

Adjacent sequences: A290313 A290314 A290315 * A290317 A290318 A290319

Keyword

nonn,tabl

Author

Wolfdieter Lang, Aug 08 2017

Status

approved