A290315 - OEIS

%I #16 Jan 25 2020 20:54:31

%S 1,1,2,1,16,12,1,66,284,120,1,224,2872,5952,1680,1,706,21080,116336,

%T 146064,30240,1,2160,132228,1531072,4804656,4130304,665280,1,6530,

%U 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

%N 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.

%C 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.

%C 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).

%C 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.

%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], 2017.

%F 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.

%e The triangle T(n, k) begins:

%e n\k  0    1      2        3         4         5         6        7 ...

%e 0:   1

%e 1:   1    2

%e 2:   1   16     12

%e 3:   1   66    284      120

%e 4:   1  224   2872     5952      1680

%e 5:   1  706  21080   116336    146064     30240

%e 6:   1 2160 132228  1531072   4804656   4130304    665280

%e 7:   1 6530 760500 16271080 101422640 208791648 132557760 17297280

%e ...

%e n = 8: 1 19648 4155120 151922560 1661273440 6556459008 9657333504 4766423040 518918400,

%e n = 9: 1 59010 21993776 1304454880 23155279200 155184721088 427142449920 477104352768 189945688320 17643225600.

%e ...

%e 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.

%Y Cf. A154537, A290311.

%K nonn,tabl

%O 0,3

%A _Wolfdieter Lang_, Jul 29 2017