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

Author

Roger L. Bagula, Mar 07 2008

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

Cf. A000045, A122525, A137216.

Programs

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 *)

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

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).

Extensions

Edited by G. C. Greubel, Jan 06 2022