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

0,6

References

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

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A000166, A000240, A000312, A137341.

Sequence in context: A290328 A200682 A263618 * A130801 A280579 A086165

Adjacent sequences: A156785 A156786 A156787 * A156789 A156790 A156791

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 15 2009

Extensions

Edited by G. C. Greubel, Jun 10 2021

Status

approved