A222523 - OEIS

%I #7 Jan 15 2025 19:54:51

%S 1,2,16,208,3930,97956,3038968,112911296,4889301222,241822567180,

%T 13450863716048,831128810329632,56483233790927556,4187162929534240488,

%U 336244786874092579920,29077531985735270053632,2694076376135933879002566,266245292488900189811625612,27956094249950913890814701248

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

%F a(n) = 1/n! * [x^n] Sum_{k>=0} (k^2+k+1)^k * x^k / (1 + (k^2+k+1)*x)^(k+1).

%F a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (k^2+k+1)^n.

%e O.g.f.: A(x) = 1 + 2*x + 16*x^2 + 208*x^3 + 3930*x^4 + 97956*x^5 +...

%e where

%e A(x) = exp(-x) + 3*x*exp(-3*x) + 7^2*exp(-7*x)*x^2/2! + 13^3*exp(-13*x)*x^3/3! + 21^4*exp(-21*x)*x^4/4! + 31^5*exp(-31*x)*x^5/5! +...

%e is a power series in x with integer coefficients.

%o (PARI) {a(n)=polcoeff(sum(k=0, n, (k^2+k+1)^k*exp(-(k^2+k+1)*x +x*O(x^n))*x^k/k!), n)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, (k^2+k+1)^k*x^k/(1+(k^2+k+1)*x +x*O(x^n))^(k+1)), n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(k^2+k+1)^n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A217900, A217901.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 24 2013