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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.

%I A137227 #8 Jan 06 2022 02:30:32
%S A137227 1,1,1,2,2,2,6,9,9,22,24,64,64,266,708,120,625,625,4536,17457,108129,
%T A137227 720,7776,7776,100392,563088,5709120,52517688,5040,117649,117649,
%U A137227 2739472,22516209,375217945,5489293264,92757410569,40320,2097152,2097152,89020752,1076444064,29566405440,688833593904,18867973329344,513683908057152
%N 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.
%H A137227 G. C. Greubel, <a href="/A137227/b137227.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A137227 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).
%e A137227 Triangle begins as:
%e A137227      1;
%e A137227      1,      1;
%e A137227      2,      2,      2;
%e A137227      6,      9,      9,      22;
%e A137227     24,     64,     64,     266,      708;
%e A137227    120,    625,    625,    4536,    17457,    108129;
%e A137227    720,   7776,   7776,  100392,   563088,   5709120,   52517688;
%e A137227   5040, 117649, 117649, 2739472, 22516209, 375217945, 5489293264, 92757410569;
%t A137227 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}])]];
%t A137227 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 06 2022 *)
%o A137227 (Sage)
%o A137227 @CachedFunction
%o A137227 def A137227(n,k):
%o A137227     if (k==0): return factorial(n)
%o A137227     elif (k==1): return n^(n-1)
%o A137227     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)))
%o A137227 flatten([[A137227(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jan 06 2022
%Y A137227 Cf. A000045, A122525, A137216.
%K A137227 nonn,tabl
%O A137227 0,4
%A A137227 _Roger L. Bagula_, Mar 07 2008
%E A137227 Edited by _G. C. Greubel_, Jan 06 2022