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
Examples
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.
Links
- 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.
Comments