%I #9 Jul 10 2018 08:01:22

%S 1,10,78,561,3885,26565,180285,1221554,8272252,56063900,380361212,

%T 2583878630,17575724491,119705606020,816297170565,5572946307857,

%U 38088275031435,260576838539320,1784382167211378,12229792806897910,83888652677221112,575858960208964685,3955813057814040153,27192049709537787123,187032147327469550926,1287187641890879422980,8863461073824746853534,61064188079233277265138,420899733623010047381885,2902469328540659624278455

%N a(n) equals the coefficient of x^n in Sum_{m>=0} (x^m + 5 + 1/x^m)^m for n >= 1.

%C The coefficient of 1/x^n in Sum_{m>=0} (x^m + 5 + 1/x^m)^m equals a(n) for n > 0, while the constant term in the sum increases without limit.

%C a(n) = Sum_{k=0..n-1} A316590(n,k) * 5^k for n >= 1.

%H Paul D. Hanna, <a href="/A316595/b316595.txt">Table of n, a(n) for n = 1..260</a>

%F a(n) ~ 7^(n + 1/2) / (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 10 2018

%e G.f.: A(x) = x + 10*x^2 + 78*x^3 + 561*x^4 + 3885*x^5 + 26565*x^6 + 180285*x^7 + 1221554*x^8 + 8272252*x^9 + 56063900*x^10 + ...

%e such that Sum_{m>=0} (x^m + 5 + 1/x^m)^m = A(x) + A(1/x) + (infinity)*x^0.

%o (PARI) {a(n) = polcoeff( sum(m=1,n, (x^-m + 5 + x^m)^m +x*O(x^n)), n,x)}

%o for(n=1,40, print1(a(n),", "))

%Y Cf. A304638, A316590, A316591, A316592, A316593, A316594.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 08 2018