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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, 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
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.
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
4: 1 224 2872 5952 1680
5: 1 706 21080 116336 146064 30240
6: 1 2160 132228 1531072 4804656 4130304 665280
7: 1 6530 760500 16271080 101422640 208791648 132557760 17297280
...
n = 8: 1 19648 4155120 151922560 1661273440 6556459008 9657333504 4766423040 518918400,
n = 9: 1 59010 21993776 1304454880 23155279200 155184721088 427142449920 477104352768 189945688320 17643225600.
...
n=3: The o.g.f. of the 4th diagonal sequence of A154537, [1, 80, 1320, ...], is P(3, x) = (1 + 66*x + 284*x^2 + 120*x^3)/(1 - 2*x)^7.
CROSSREFS
Sequence in context: A324610 A373034 A247125 * A113108 A162005 A325220
KEYWORD
nonn,tabl
AUTHOR
Wolfdieter Lang, Jul 29 2017
STATUS
approved