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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 Stirling 2), 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 (list; table; graph; refs; listen; history; text; internal format)
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.

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 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 4-th 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

Cf. A154537, A290311.

Sequence in context: A016447 A095850 A247125 * A113108 A162005 A013125

Adjacent sequences:  A290312 A290313 A290314 * A290316 A290317 A290318

KEYWORD

nonn,tabl

AUTHOR

Wolfdieter Lang, Jul 29 2017

STATUS

approved

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Last modified October 20 07:17 EDT 2018. Contains 316378 sequences. (Running on oeis4.)