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

%I

%S 1,8,300,87808,226060880,5189199673344,1059339584023000000,

%T 1920530508944313150734336,30940855519482868628911503608064,

%U 4436292850480330161986749628748086640640,5670112877213289113061309258575292968750000000000,64696388003897763308476146732369244462492297181932629786624

%N a(n) = (n+1) * (3^n + 1)^n.

%C More generally, the following sums are equal:

%C (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

%C (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),

%C for any fixed integer k; here, k = 2 and p = 1, q = 3, r = x.

%F O.g.f.: Sum_{n>=0} (n+1) * (3^n + 1)^n * x^n.

%F O.g.f.: Sum_{n>=0} (n+1) * 3^(n^2) * x^n / (1 - 3^n*x)^(n+2).

%F E.g.f.: sum_{n>=0} (n+1 + 3^n*x) * 3^(n^2) * exp(3^n*x) * x^n/n!.

%e 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 such that

%e 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 (PARI) {a(n) = (n+1) * (3^n + 1)^n}

%o for(n=0,15, print1(a(n),", "))

%o (PARI) /* O.g.f. */

%o {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 for(n=0,15, print1(a(n),", "))

%o (PARI) /* E.g.f. */

%o {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 for(n=0,15, print1(a(n),", "))

%Y Cf. A202989.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 05 2019

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Last modified February 25 08:19 EST 2020. Contains 332221 sequences. (Running on oeis4.)