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, 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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Adjacent sequences: A155905 A155906 A155907 * A155909 A155910 A155911

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 30 2009

Extensions

Edited by G. C. Greubel, Jun 05 2021

Status

approved