A293609 Triangle read by rows, a refinement of the central Stirling numbers of the first kind A187646, T(n, k) for n >= 0 and 0 <= k <= n.

1, 1, 0, 7, 4, 0, 90, 120, 15, 0, 1701, 3696, 1316, 56, 0, 42525, 129780, 84630, 12180, 210, 0, 1323652, 5233404, 5184894, 1492744, 104049, 792, 0, 49329280, 240240000, 326426100, 151251100, 22840818, 852852, 3003, 0

Offset: 0

Author

Peter Luschny, Oct 15 2017

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]        1
[1]        1,         0
[2]        7,         4,         0
[3]       90,       120,        15,         0
[4]     1701,      3696,      1316,        56,        0
[5]    42525,    129780,     84630,     12180,      210,      0
[6]  1323652,   5233404,   5184894,   1492744,   104049,    792,    0
[7] 49329280, 240240000, 326426100, 151251100, 22840818, 852852, 3003, 0
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Row sums are A187646. T(n, 0) = A007820(n) the central Stirling numbers of the second kind A048993. T(n, n-1) = A001791(n) for n>=1.

Cf. A293616.

Programs

Maple

for n in [$0..9] do seq(A293616(n, n, k), k=0..n) od;

Mathematica

A293609Row[n_] := If[n==0, {1}, Join[CoefficientList[x^(-n) (1 - x)^(2n) PolyLog[-2n, n, x] /. Log[1 - x] -> 0, x], {0}]];
Table[A293609Row[n], {n, 0, 7}] // Flatten

Formula

T(n, k) = A293616(n, n, k) for k = 0..n. The main diagonal in terms of rows (!) of the array of triangles A293616. T_row(n) is row n of triangle A293616(n,.,.), i.e. T_row(0) = [1] is row 0 of A000007, T_row(1) = [1, 0] is row 1 of A173018, T_row(2) = [7, 4, 0] is row 2 of A062253, and so on.

Let h(n) = x^(-n)*(1 - x)^(2*n)*PolyLog(-2*n, n, x) and p(n) the polynomial given by the expansion of h(n) after replacing log(1 - x) by 0. Then T(n, k) is the k-th coefficient of p(n) for 0 <= k < n.