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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.
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%I #15 Oct 23 2017 05:26:18

%S 1,1,0,7,4,0,90,120,15,0,1701,3696,1316,56,0,42525,129780,84630,12180,

%T 210,0,1323652,5233404,5184894,1492744,104049,792,0,49329280,

%U 240240000,326426100,151251100,22840818,852852,3003,0

%N 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 G. C. Greubel, <a href="/A293609/b293609.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F 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 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 Triangle starts:

%e [0] 1

%e [1] 1, 0

%e [2] 7, 4, 0

%e [3] 90, 120, 15, 0

%e [4] 1701, 3696, 1316, 56, 0

%e [5] 42525, 129780, 84630, 12180, 210, 0

%e [6] 1323652, 5233404, 5184894, 1492744, 104049, 792, 0

%e [7] 49329280, 240240000, 326426100, 151251100, 22840818, 852852, 3003, 0

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

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

%t Table[A293609Row[n], {n, 0, 7}] // Flatten

%Y 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 Cf. A293616.

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Oct 15 2017