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A155868 Triangle T(n, k) = (-1)^n*StirlingS1(n, j)*StirlingS1(n, n-j), with T(n, 0) = T(n, n) = 1, read by rows.

%I A155868 #7 Sep 08 2022 08:45:41
%S A155868 1,1,1,1,1,1,1,6,6,1,1,36,121,36,1,1,240,1750,1750,240,1,1,1800,23290,
%T A155868 50625,23290,1800,1,1,15120,308700,1193640,1193640,308700,15120,1,1,
%U A155868 141120,4207896,25738720,45819361,25738720,4207896,141120,1,1,1451520,59832864,535810464,1510458516,1510458516,535810464,59832864,1451520,1
%N A155868 Triangle T(n, k) = (-1)^n*StirlingS1(n, j)*StirlingS1(n, n-j), with T(n, 0) = T(n, n) = 1, read by rows.
%C A155868 Row sums are:
%C A155868 {1, 2, 3, 14, 195, 3982, 100807, 3034922, 105994835, 4215106730, 188097696347,...}
%H A155868 G. C. Greubel, <a href="/A155868/b155868.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155868 T(n, k) = coefficients of p(n, x), where p(n, x) = 1 + x^n + (-1)^n*Sum_{j=0..n} StirlingS1(n, j)*StirlingS1(n, n-j)*x^k and p(0, x) = 1.
%F A155868 From _G. C. Greubel_, Jun 04 2021: (Start)
%F A155868 T(n, k) = (-1)^n*StirlingS1(n, j)*StirlingS1(n, n-j), with T(n, 0) = T(n, n) = 1.
%F A155868 Sum_{k=0..n} T(n, k) = 2 + A342111(n) - 2*[n==0]. (End)
%e A155868 Triangle begins as:
%e A155868   1;
%e A155868   1,      1;
%e A155868   1,      1,       1;
%e A155868   1,      6,       6,        1;
%e A155868   1,     36,     121,       36,        1;
%e A155868   1,    240,    1750,     1750,      240,        1;
%e A155868   1,   1800,   23290,    50625,    23290,     1800,       1;
%e A155868   1,  15120,  308700,  1193640,  1193640,   308700,   15120,      1;
%e A155868   1, 141120, 4207896, 25738720, 45819361, 25738720, 4207896, 141120, 1;
%t A155868 (* First program *)
%t A155868 p[n_, x_]:= If[n==0, 1, 1 +x^n +(-1)^n*Sum[StirlingS1[n, j]*StirlingS1[n, n-j]*x^j, {j,0,n}]];
%t A155868 Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten (* modified by _G. C. Greubel_, Jun 04 2021 *)
%t A155868 (* Second program *)
%t A155868 T[n_, k_]:=  If[k==0 || k==n, 1, (-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k]];
%t A155868 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 04 2021 *)
%o A155868 (Magma)
%o A155868 A155868:= func< n,k | k eq 0 or k eq n select 1 else (-1)^n*StirlingFirst(n,k)* StirlingFirst(n,n-k) >;
%o A155868 [A155868(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 04 2021
%o A155868 (Sage)
%o A155868 def A155868(n,k): return 1 if (k==0 or k==n) else stirling_number1(n,k)*stirling_number1(n,n-k)
%o A155868 flatten([[A155868(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 04 2021
%Y A155868 Cf. A048994, A342111.
%K A155868 nonn,tabl
%O A155868 0,8
%A A155868 _Roger L. Bagula_, Jan 29 2009
%E A155868 Edited by _G. C. Greubel_, Jun 04 2021