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

%I A181547 #18 Apr 23 2025 16:19:52
%S A181547 1,1,2,33,245,1268,10903,108801,876184,7319995,70550669,663827604,
%T A181547 6051592703,57695451167,563736086740,5452227384417,53094611797387,
%U A181547 525962074892014,5232943624317191,52145361057635835,523458523860890906
%N A181547 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^5.
%C A181547 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).
%C A181547 For this sequence (L=5): lim_{n->oo} a(n+1)/a(n) = (5*sqrt(5)+11)/2 = 11.090...
%C A181547 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
%H A181547 Seiichi Manyama, <a href="/A181547/b181547.txt">Table of n, a(n) for n = 0..963</a>
%e A181547 G.f. A(x) = 1 + x + 2*x^2 + 33*x^3 + 245*x^4 + 1268*x^5 + 10903*x^6 +...
%e A181547 The terms begin:
%e A181547 a(0) = a(1) = 1^5;
%e A181547 a(2) = 1^5 + 1^5 = 2;
%e A181547 a(3) = 1^5 + 2^5 = 33;
%e A181547 a(4) = 1^5 + 3^5 + 1^5 = 245;
%e A181547 a(5) = 1^5 + 4^5 + 3^5 = 1268;
%e A181547 a(6) = 1^5 + 5^5 + 6^5 + 1^5 = 10903;
%e A181547 a(7) = 1^5 + 6^5 + 10^5 + 4^5 = 108801; ...
%o A181547 (PARI) {a(n)=sum(k=0,n\2,binomial(n-k,k)^5)}
%Y A181547 Cf. variants: A181545, A181546, A051286.
%Y A181547 Cf. A000032, A000045.
%K A181547 nonn
%O A181547 0,3
%A A181547 _Paul D. Hanna_, Oct 29 2010