A156788 Triangle T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1, read by rows.

1, 0, 1, 0, 0, 4, 0, 3, 0, 27, 0, 8, 96, 0, 256, 0, 45, 640, 2430, 0, 3125, 0, 264, 8640, 29160, 61440, 0, 46656, 0, 1855, 118272, 688905, 1146880, 1640625, 0, 823543, 0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248, 0, 16777216, 0, 133497, 34172928, 438143580, 1453326336, 2214843750, 1693052928, 1452729852, 0, 387420489

Offset: 0

Author

Roger L. Bagula, Feb 15 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  0,     1;
  0,     0,       4;
  0,     3,       0,       27;
  0,     8,      96,        0,      256;
  0,    45,     640,     2430,        0,     3125;
  0,   264,    8640,    29160,    61440,        0,    46656;
  0,  1855,  118272,   688905,  1146880,  1640625,        0, 823543;
  0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248,      0, 16777216;
		

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p.194.

Links

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

Crossrefs

Cf. A000166, A000240, A000312, A137341.

Programs

Mathematica

A000166[n_]:= A000166[n]= If[n==0, 1, n*A000166[n-1] + (-1)^n];
T[n_, k_]:= If[n==0, 1, Binomial[n, k]*A000166[n-k]*k^n];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 10 2021 *)

Sage

def A000166(n): return 1 if (n==0) else n*A000166(n-1) + (-1)^n
def A156788(n,k): return 1 if (n==0) else binomial(n,k)*k^n*A000166(n-k)
flatten([[A156788(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 10 2021

Formula

T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1.

T(n, k) = binomial(n, k)*b(n-k)*k^n, where b(n) = n*b(n-1) + (-1)^n and b(0) = 1.

Sum_{k=0..n} T(n, k) = A137341(n).

From G. C. Greubel, Jun 10 2021: (Start)

T(n, 1) = A000240(n).

T(n, n) = A000312(n). (End)

Extensions

Edited by G. C. Greubel, Jun 10 2021