A193421 E.g.f.: Sum_{n>=0} x^n * exp(n^2*x).

1, 1, 4, 33, 436, 8185, 206046, 6622945, 263313688, 12627149265, 716160702970, 47284266221401, 3587061106583604, 309251317536586633, 30017652739792964806, 3254137305364883664945, 391238883136463492841136, 51846176797206158144925985

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..290

Formula

E.g.f.: A(x) = Sum_{n>=0} x^n*exp(n*x)*Product_{k=1..n} (1 - x*exp((4*k-3)*x)) / (1 - x*exp((4*k-1)*x)), due to a q-series identity.

Let q = exp(x), then the e.g.f. equals the continued fraction:

A(x) = 1/(1- q*x/(1- q*(q^2-1)*x/(1- q^5*x/(1- q^3*(q^4-1)*x/(1- q^9*x/(1- q^5*(q^6-1)*x/(1- q^13*x/(1- q^7*(q^8-1)*x/(1- ...))))))))), due to a partial elliptic theta function identity.

a(n) = n! * Sum_{k=0..n} (n-k)^(2*k)/k!. - Paul D. Hanna, Jan 19 2013

O.g.f.: Sum_{k>=0} k! * x^k / (1 - k^2*x)^(k+1). - Ilya Gutkovskiy, Jul 02 2019

log(a(n)) ~ n*(2*(log(n) - 1) + LambertW(sqrt(n))*(3*log(n) - 2*log(1 + LambertW(sqrt(n))) + 2*LambertW(sqrt(n)))) / (2*(1 + LambertW(sqrt(n)))). - Vaclav Kotesovec, Nov 26 2022

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 436*x^4/4! + 8185*x^5/5! + 206046*x^6/6! +...
where
A(x) = 1 + x*exp(x) + x^2*exp(4*x) + x^3*exp(9*x) + x^4*exp(16*x) +...
By a q-series identity:
A(x) = 1 + x*exp(x)*(1-x*exp(x))/(1-x*exp(3*x)) + x^2*exp(2*x)*(1-x*exp(x))*(1-x*exp(5*x))/((1-x*exp(3*x))*(1-x*exp(7*x))) + x^3*exp(3*x)*(1-x*exp(x))*(1-x*exp(5*x))*(1-x*exp(9*x))/((1-x*exp(3*x))*(1-x*exp(7*x))*(1-x*exp(11*x))) +...

Mathematica

Flatten[{1, Table[n! * Sum[(n-k)^(2*k)/k!, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 21 2014 *)

Prog

(PARI) {a(n)=local(Egf); Egf=sum(m=0, n, x^m*exp(m^2*x+x*O(x^n))); n!*polcoeff(Egf, n)}
(PARI) /* q-series identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*exp(m*x+x*O(x^n))*prod(k=1, m, (1-x*exp((4*k-3)*x+x*O(x^n)))/(1-x*exp((4*k-1)*x+x*O(x^n)))))); n!*polcoeff(A, n)}
(PARI) {a(n) = n!*sum(k=0, n, (n-k)^(2*k)/k!)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A006153, A193467.

Sequence in context: A119821 A102321 A268293 * A179421 A356834 A296835

Adjacent sequences: A193418 A193419 A193420 * A193422 A193423 A193424

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2011

Status

approved