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a(n) equals the coefficient of x^n in Sum_{m>=0} (x^m + 3 + 1/x^m)^m for n >= 1.
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%I #10 Jul 10 2018 07:59:04

%S 1,6,30,145,685,3267,15533,74338,356284,1714020,8263596,39940398,

%T 193419915,938440188,4560542645,22196008209,108171753355,527816934216,

%U 2578310320610,12607506013260,61706212041096,302275147959675,1481908332595625,7270432038855843,35694090764454926,175351391301452028,861946790726717742,4239292356515821416,20860889073855326397,102703447427882985153

%N a(n) equals the coefficient of x^n in Sum_{m>=0} (x^m + 3 + 1/x^m)^m for n >= 1.

%C The coefficient of 1/x^n in Sum_{m>=0} (x^m + 3 + 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) * 3^k for n >= 1.

%H Paul D. Hanna, <a href="/A316593/b316593.txt">Table of n, a(n) for n = 1..300</a>

%F a(n) ~ 5^(n + 1/2) / (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 10 2018

%e G.f.: A(x) = x + 6*x^2 + 30*x^3 + 145*x^4 + 685*x^5 + 3267*x^6 + 15533*x^7 + 74338*x^8 + 356284*x^9 + 1714020*x^10 + 8263596*x^11 + 39940398*x^12 + ...

%e such that Sum_{m>=0} (x^m + 3 + 1/x^m)^m = A(x) + A(1/x) + (infinity)*x^0.

%o (PARI) {a(n) = polcoeff( sum(m=1,n, (x^-m + 3 + x^m)^m +x*O(x^n)), n,x)}

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

%Y Cf. A304638, A316590, A316591, A316592, A316594, A316595.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 08 2018