login
A155908
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.
1
1, 1, 1, 1, 6, 1, 1, 27, 27, 1, 1, 156, 262, 156, 1, 1, 1375, 2560, 2560, 1375, 1, 1, 16998, 33303, 34052, 33303, 16998, 1, 1, 262591, 576261, 546875, 546875, 576261, 262591, 1, 1, 4783992, 12054460, 11922248, 9222918, 11922248, 12054460, 4783992, 1
OFFSET
0,5
REFERENCES
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 60
FORMULA
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.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 27, 27, 1;
1, 156, 262, 156, 1;
1, 1375, 2560, 2560, 1375, 1;
1, 16998, 33303, 34052, 33303, 16998, 1;
1, 262591, 576261, 546875, 546875, 576261, 262591, 1;
1, 4783992, 12054460, 11922248, 9222918, 11922248, 12054460, 4783992, 1;
MATHEMATICA
f[n_, k_]:= Sum[(-1)^(k-j+1)*Binomial[n+1, j]*j^n, {j, 0, k+1}]/(n+1);
T[n_, k_]:= If[n==0, 1, f[n, k] + f[n, n-k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 05 2021 *)
PROG
(Sage)
def f(n, k): return sum( (-1)^(k+j+1)*j^n*binomial(n+1, j) for j in (0..k+1) )
def A155908(n, k): return 1 if (n==0) else f(n, k) + f(n, n-k)
flatten([[A155908(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021
CROSSREFS
Sequence in context: A140945 A141688 A166960 * A105373 A296548 A201461
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 30 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 05 2021
STATUS
approved