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A156788
Triangle T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1, read by rows.
1
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.
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 15 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 10 2021
STATUS
approved