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%I #7 Dec 26 2023 12:32:25
%S 1,1,1,1,6,1,1,27,27,1,1,156,262,156,1,1,1375,2560,2560,1375,1,1,
%T 16998,33303,34052,33303,16998,1,1,262591,576261,546875,546875,576261,
%U 262591,1,1,4783992,12054460,11922248,9222918,11922248,12054460,4783992,1
%N Triangle T(n, k) = f(n, k) + f(n, n-k) where T(0, 0) = 1 and f(n, k) = 1/(n+1)*Sum_{j=0..k+1} (-1)^(k-j+1)* binomial(n+1, j)*j^n, read by rows.
%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 60
%H G. C. Greubel, <a href="/A155908/b155908.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = f(n, k) + f(n, n-k) with T(0, 0) = 1, where f(n, k) = 1/(n+1)*Sum_{j=0..k+1} (-1)^(k-j+1)* binomial(n+1, j)*j^n.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 27, 27, 1;
%e 1, 156, 262, 156, 1;
%e 1, 1375, 2560, 2560, 1375, 1;
%e 1, 16998, 33303, 34052, 33303, 16998, 1;
%e 1, 262591, 576261, 546875, 546875, 576261, 262591, 1;
%e 1, 4783992, 12054460, 11922248, 9222918, 11922248, 12054460, 4783992, 1;
%t f[n_, k_]:= Sum[(-1)^(k-j+1)*Binomial[n+1, j]*j^n, {j, 0, k+1}]/(n+1);
%t T[n_, k_]:= If[n==0, 1, f[n, k] + f[n, n-k]];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 05 2021 *)
%o (Sage)
%o def f(n,k): return sum( (-1)^(k+j+1)*j^n*binomial(n+1,j) for j in (0..k+1) )
%o def A155908(n,k): return 1 if (n==0) else f(n,k) + f(n,n-k)
%o flatten([[A155908(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 05 2021
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Jan 30 2009
%E Edited by _G. C. Greubel_, Jun 05 2021
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