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A326013 a(n) = (n+1) * (3^n + 1)^n.

%I A326013 #5 Jul 07 2019 09:51:52
%S A326013 1,8,300,87808,226060880,5189199673344,1059339584023000000,
%T A326013 1920530508944313150734336,30940855519482868628911503608064,
%U A326013 4436292850480330161986749628748086640640,5670112877213289113061309258575292968750000000000,64696388003897763308476146732369244462492297181932629786624
%N A326013 a(n) = (n+1) * (3^n + 1)^n.
%C A326013 More generally, the following sums are equal:
%C A326013 (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C A326013 (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
%C A326013 for any fixed integer k; here, k = 2 and p = 1, q = 3, r = x.
%F A326013 O.g.f.: Sum_{n>=0} (n+1) * (3^n + 1)^n * x^n.
%F A326013 O.g.f.: Sum_{n>=0} (n+1) * 3^(n^2) * x^n / (1 - 3^n*x)^(n+2).
%F A326013 E.g.f.: sum_{n>=0} (n+1 + 3^n*x) * 3^(n^2) * exp(3^n*x) * x^n/n!.
%e A326013 O.g.f.: A(x) = 1 + 8*x + 300*x^2 + 87808*x^3 + 226060880*x^4 + 5189199673344*x^5 + 1059339584023000000*x^6 + 1920530508944313150734336*x^7 + ... + (n+1) * (3^n + 1)^n*x^n + ...
%e A326013 such that
%e A326013 A(x) = 1/(1 - x)^2 + 2*3*x/(1 - 3*x)^3 + 3*3^4*x^2/(1 - 3^2*x)^4 + 4*3^9*x^3/(1 - 3^3*x)^5 + 5*3^16*x^4/(1 - 3^4*x)^6 + 6*3^25*x^5/(1 - 3^5*x)^7 + 7*3^36*x^6/(1 - 3^6*x)^8 + ... (n+1)*3^(n^2)*x^n/(1 - 3^n*x)^(n+2) + ...
%o A326013 (PARI) {a(n) = (n+1) * (3^n + 1)^n}
%o A326013 for(n=0,15, print1(a(n),", "))
%o A326013 (PARI) /* O.g.f. */
%o A326013 {a(n) = my(A = sum(m=0,n, (m+1) * 3^(m^2) * x^m / (1 - 3^m*x +x*O(x^n))^(m+2) )); polcoeff(A,n)}
%o A326013 for(n=0,15, print1(a(n),", "))
%o A326013 (PARI) /* E.g.f. */
%o A326013 {a(n) = my(A = sum(m=0,n, (m+1 + 3^m*x) * 3^(m^2) * exp(3^m*x +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
%o A326013 for(n=0,15, print1(a(n),", "))
%Y A326013 Cf. A202989.
%K A326013 nonn
%O A326013 0,2
%A A326013 _Paul D. Hanna_, Jun 05 2019