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
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
KEYWORD
nonn,tabl
AUTHOR
Wolfdieter Lang, Aug 08 2017
STATUS
approved