A181547 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^5.

1, 1, 2, 33, 245, 1268, 10903, 108801, 876184, 7319995, 70550669, 663827604, 6051592703, 57695451167, 563736086740, 5452227384417, 53094611797387, 525962074892014, 5232943624317191, 52145361057635835, 523458523860890906

Offset: 0

Author

Paul D. Hanna, Oct 29 2010

Keywords

• nonn

Comments

Conjecture: Given F(n,L) = Sum_{k=0..[n/2]} C(n-k,k)^L, then lim_{n->oo} F(n+1,L)/F(n,L) = (Fibonacci(L)*sqrt(5) + Lucas(L))/2 for L>=0 where Fibonacci(n) = A000045(n) and Lucas(n) = A000032(n).

For this sequence (L=5): lim_{n->oo} a(n+1)/a(n) = (5*sqrt(5)+11)/2 = 11.090...

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - u)*(1 - v) - (x*y*z*u*v)^2). - Ilya Gutkovskiy, Apr 23 2025

Examples

			G.f. A(x) = 1 + x + 2*x^2 + 33*x^3 + 245*x^4 + 1268*x^5 + 10903*x^6 +...
The terms begin:
a(0) = a(1) = 1^5;
a(2) = 1^5 + 1^5 = 2;
a(3) = 1^5 + 2^5 = 33;
a(4) = 1^5 + 3^5 + 1^5 = 245;
a(5) = 1^5 + 4^5 + 3^5 = 1268;
a(6) = 1^5 + 5^5 + 6^5 + 1^5 = 10903;
a(7) = 1^5 + 6^5 + 10^5 + 4^5 = 108801; ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..963

Crossrefs

Cf. variants: A181545, A181546, A051286.

Cf. A000032, A000045.

Programs

PARI

{a(n)=sum(k=0,n\2,binomial(n-k,k)^5)}