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A181546 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^4.

%I A181546 #24 Apr 23 2025 16:19:47
%S A181546 1,1,2,17,83,338,1923,11553,63028,359203,2172469,13026034,78106885,
%T A181546 478415635,2957675956,18321372721,114301292581,718253640196,
%U A181546 4531427831111,28699590926291,182566373639352,1165539703613397
%N A181546 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^4.
%C A181546 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 A181546 For this sequence (L=4): lim_{n->oo} a(n+1)/a(n) = (3*sqrt(5)+7)/2 = 6.8541...
%C A181546 Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y*z*w)^2). - _Ilya Gutkovskiy_, Apr 23 2025
%H A181546 Seiichi Manyama, <a href="/A181546/b181546.txt">Table of n, a(n) for n = 0..1202</a>
%H A181546 C. Banderier, P. Hitczenko, <a href="https://doi.org/10.1016/j.dam.2011.12.011">Enumeration and asymptotics of restricted compositions having the same number of parts</a>, Disc. Appl. Math. 160 (18) (2012) 2542-2554. Table 1.
%e A181546 G.f. A(x) = 1 + x + 2*x^2 + 17*x^3 + 83*x^4 + 338*x^5 + 1923*x^6 +...
%e A181546 The terms begin:
%e A181546 a(0) = a(1) = 1^4;
%e A181546 a(2) = 1^4 + 1^4 = 2;
%e A181546 a(3) = 1^4 + 2^4 = 17;
%e A181546 a(4) = 1^4 + 3^4 + 1^4 = 83;
%e A181546 a(5) = 1^4 + 4^4 + 3^4 = 338;
%e A181546 a(6) = 1^4 + 5^4 + 6^4 + 1^4 = 1923;
%e A181546 a(7) = 1^4 + 6^4 + 10^4 + 4^4 = 11553; ...
%t A181546 Table[Sum[Binomial[n-k,k]^4,{k,0,Floor[n/2]}],{n,0,30}] (* _Harvey P. Dale_, May 22 2021 *)
%o A181546 (PARI) {a(n)=sum(k=0,n\2,binomial(n-k,k)^4)}
%Y A181546 Cf. variants: A181545, A181547, A051286.
%Y A181546 Cf. A000032, A000045.
%K A181546 nonn
%O A181546 0,3
%A A181546 _Paul D. Hanna_, Oct 29 2010