A155908 - OEIS

%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