A320830 - OEIS

%I #16 Dec 15 2024 04:35:24

%S 1,2,6,30,504,11530,250800,7239974,316070272,15395970258,814193153280,

%T 53776871363182,4193637641554944,353769117203810906,

%U 33739767209254672384,3696794844865401890550,443059728692677637406720,57958478772549018401984674,8422823140216886323795525632,1335371306389226812255794229694

%N E.g.f.: [Sum_{n>=0} x^n * exp(n^2*x)] * [Sum_{n>=0} x^n / exp(n^2*x)].

%H Paul D. Hanna, <a href="/A320830/b320830.txt">Table of n, a(n) for n = 0..300</a>

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

%e E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 30*x^3/3! + 504*x^4/4! + 11530*x^5/5! + 250800*x^6/6! + 7239974*x^7/7! + 316070272*x^8/8! + 15395970258*x^9/9! + ...

%e such that A(x) = P(x) * Q(x) where

%e P(x) = 1 + x*exp(x) + x^2*exp(4*x) + x^3*exp(9*x) + x^4*exp(16*x) + x^5*exp(25*x) + x^6*exp(36*x) + x^7*exp(49*x) + ... + x^n * exp(n^2*x) + ...

%e Q(x) = 1 + x/exp(x) + x^2/exp(4*x) + x^3/exp(9*x) + x^4/exp(16*x) + x^5/exp(25*x) + x^6/exp(36*x) + x^7/exp(49*x) + ... + x^n / exp(n^2*x) + ...

%e Explicitly,

%e P(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 436*x^4/4! + 8185*x^5/5! + 206046*x^6/6! + 6622945*x^7/7! + 263313688*x^8/8! + ... + A193421(n)*x^n/n! + ...

%e Q(x) = 1 + x - 15*x^3/3! - 4*x^4/4! + 1785*x^5/5! - 4926*x^6/6! - 707231*x^7/7! + 9681384*x^8/8! + 593043921*x^9/9! + ...

%e RELATED SERIES.

%e log(A(x)) = 2*x + 2*x^2/2! + 10*x^3/3! + 348*x^4/4! + 7138*x^5/5! + 127440*x^6/6! + 4143914*x^7/7! + 207951968*x^8/8! + 9863732610*x^9/9! + ...

%o (PARI) {a(n) = my(A = sum(m=0,n, x^m*exp(m^2*x + x*O(x^n)) ) * sum(m=0,n, x^m*exp(-m^2*x + x*O(x^n)) )); n!*polcoeff(A,n)}

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

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

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

%Y Cf. A193421, A320950.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 22 2018