A137227 Triangle T(n, k) = n^(n-1) * Fibonacci(k)^(n+1) - (n-1)! * (Fibonacci(k) - 1) * Sum_{j=0..n} (n*Fibonacci(k))^j/j!, with T(n, 0) = n! and T(n, 1) = n^(n-1), read by rows.

1, 1, 1, 2, 2, 2, 6, 9, 9, 22, 24, 64, 64, 266, 708, 120, 625, 625, 4536, 17457, 108129, 720, 7776, 7776, 100392, 563088, 5709120, 52517688, 5040, 117649, 117649, 2739472, 22516209, 375217945, 5489293264, 92757410569, 40320, 2097152, 2097152, 89020752, 1076444064, 29566405440, 688833593904, 18867973329344, 513683908057152

Offset

0,4

Links

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

Formula

T(n, k) = (1/n)*( n^n * Fibonacci(k)^(n+1) - n! * (Fibonacci(k) - 1) * Sum_{j=0..n} (n*Fibonacci(k))^j/j! ), with T(n, 0) = n! and T(n, 1) = n^(n-1).

Example

Triangle begins as:
     1;
     1,      1;
     2,      2,      2;
     6,      9,      9,      22;
    24,     64,     64,     266,      708;
   120,    625,    625,    4536,    17457,    108129;
   720,   7776,   7776,  100392,   563088,   5709120,   52517688;
  5040, 117649, 117649, 2739472, 22516209, 375217945, 5489293264, 92757410569;

Mathematica

T[n_, k_]:= If[k==0, n!, If[k==1, n^(n-1), (1/n)*(Fibonacci[k]^(n+1)*n^n - n!*(Fibonacci[k] -1)*Sum[n^j*Fibonacci[k]^j/j!, {j, 0, n}])]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 06 2022 *)

Prog

(Sage)
@CachedFunction
def A137227(n, k):
    if (k==0): return factorial(n)
    elif (k==1): return n^(n-1)
    else: return (1/n)*(fibonacci(k)^(n+1)*n^n - factorial(n)*(fibonacci(k) -1)*sum((n*fibonacci(k))^j/factorial(j) for j in (0..n)))
flatten([[A137227(n, k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 06 2022

Crossrefs

Cf. A000045, A122525, A137216.

Sequence in context: A323860 A121698 A087482 * A323862 A291667 A052537

Adjacent sequences: A137224 A137225 A137226 * A137228 A137229 A137230

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 07 2008

Extensions

Edited by G. C. Greubel, Jan 06 2022

Status

approved