cp's OEIS Frontend

A218679 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(4*n) * x^n/n! * exp(-n*x*(1+n*x)^4).

%I A218679 #6 Jul 25 2025 16:51:03
%S A218679 1,1,5,31,273,2652,30071,375628,5135649,75945388,1202006514,
%T A218679 20243446719,360517872287,6758311053521,132833835618576,
%U A218679 2728019848249377,58370987166092073,1297916560174624569,29924140267551540116,713934350929955200551,17594768127940813003452
%N A218679 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(4*n) * x^n/n! * exp(-n*x*(1+n*x)^4).
%C A218679 Compare o.g.f. to the curious identity:
%C A218679 1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).
%e A218679 O.g.f.: A(x) = 1 + x + 5*x^2 + 31*x^3 + 273*x^4 + 2652*x^5 + 30071*x^6 +...
%e A218679 where
%e A218679 A(x) = 1 + (1+x)^4*x*exp(-x*(1+x)^4) + 2^2*(1+2*x)^8*x^2/2!*exp(-2*x*(1+2*x)^4) + 3^3*(1+3*x)^12*x^3/3!*exp(-3*x*(1+3*x)^4) + 4^4*(1+4*x)^16*x^4/4!*exp(-4*x*(1+4*x)^4) + 5^5*(1+5*x)^20*x^5/5!*exp(-5*x*(1+5*x)^4) +...
%e A218679 simplifies to a power series in x with integer coefficients.
%o A218679 (PARI) {a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^(4*k)*x^k/k!*exp(-k*x*(1+k*x)^4+x*O(x^n)));polcoeff(A,n)}
%o A218679 for(n=0,30,print1(a(n),", "))
%Y A218679 Cf. A218670, A218677, A218678.
%K A218679 nonn
%O A218679 0,3
%A A218679 _Paul D. Hanna_, Nov 04 2012