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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.

%I A293609 #15 Oct 23 2017 05:26:18
%S A293609 1,1,0,7,4,0,90,120,15,0,1701,3696,1316,56,0,42525,129780,84630,12180,
%T A293609 210,0,1323652,5233404,5184894,1492744,104049,792,0,49329280,
%U A293609 240240000,326426100,151251100,22840818,852852,3003,0
%N 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.
%H A293609 G. C. Greubel, <a href="/A293609/b293609.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A293609 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.
%F A293609 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.
%e A293609 Triangle starts:
%e A293609 [0]        1
%e A293609 [1]        1,         0
%e A293609 [2]        7,         4,         0
%e A293609 [3]       90,       120,        15,         0
%e A293609 [4]     1701,      3696,      1316,        56,        0
%e A293609 [5]    42525,    129780,     84630,     12180,      210,      0
%e A293609 [6]  1323652,   5233404,   5184894,   1492744,   104049,    792,    0
%e A293609 [7] 49329280, 240240000, 326426100, 151251100, 22840818, 852852, 3003, 0
%p A293609 for n in [$0..9] do seq(A293616(n, n, k), k=0..n) od;
%t A293609 A293609Row[n_] := If[n==0, {1}, Join[CoefficientList[x^(-n) (1 - x)^(2n) PolyLog[-2n, n, x] /. Log[1 - x] -> 0, x], {0}]];
%t A293609 Table[A293609Row[n], {n, 0, 7}] // Flatten
%Y A293609 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.
%Y A293609 Cf. A293616.
%K A293609 nonn,tabl
%O A293609 0,4
%A A293609 _Peter Luschny_, Oct 15 2017