cp's OEIS Frontend

A202991 E.g.f: Sum_{n>=0} 3^(n^2) * exp(-2*3^n*x) * x^n/n!.

%I A202991 #8 Jan 25 2025 18:45:18
%S A202991 1,1,49,15625,38950081,812990017201,147640825624179889,
%T A202991 237771659632917369765625,3425319186561140076700951192321,
%U A202991 443021141828981570872668681812345111521,515202988063835984513918825523304657054713360049
%N A202991 E.g.f: Sum_{n>=0} 3^(n^2) * exp(-2*3^n*x) * x^n/n!.
%C A202991 E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.
%C A202991 O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.
%F A202991 a(n) = (3^n - 2)^n.
%F A202991 O.g.f.: Sum_{n>=0} 3^(n^2)*x^n/(1 + 2*3^n*x)^(n+1).
%e A202991 E.g.f.: A(x) = 1 + x + 49*x^2/2! + 15625*x^3/3! + 38950081*x^4/4! +...
%e A202991 By the series identity, the g.f.:
%e A202991 A(x) = exp(-2*x) + 3*exp(-2*3*x)*x + 3^4*exp(-2*3^2*x)*x^2/2! + 3^9*exp(-2*3^3*x)*x^3/3! + 3^16*exp(-2*3^4*x)*x^4/4! +...
%e A202991 expands into:
%e A202991 A(x) = 1 + x + 7^2*x^2/2! + 25^3*x^3/3! + 79^4*x^4/4! + 241^5*x^5/5! +...+ (3^n-2)^n*x^n/n! +...
%o A202991 (PARI) {a(n, q=3, m=1, b=-2)=(m*q^n + b)^n}
%o A202991 (PARI) {a(n, q=3, m=1, b=-2)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}
%o A202991 (PARI) {a(n, q=3, m=1, b=-2)=polcoeff(sum(k=0, n, m^k*q^(k^2)*x^k/(1-b*q^k*x+x*O(x^n))^(k+1)), n)}
%Y A202991 Cf. A180602, A165327, A202990, A202989, A060613, A055601.
%K A202991 nonn
%O A202991 0,3
%A A202991 _Paul D. Hanna_, Dec 26 2011