login
E.g.f.: Sum_{n=-oo..+oo} x^n * exp(n*x) * (exp(n*x) - x^n)^n.
1

%I #8 Oct 04 2017 20:04:04

%S 1,7,37,807,13441,413243,13468813,571503103,28308826657,1666118229819,

%T 113262531063661,8830681086125231,780324383486361793,

%U 77494753844884990123,8581533546227249944141,1052503537117606772557695,142116165804218004169556929,21014208913195247508525503483,3386598011981006005953444008269,592290509726367692126040767254639

%N E.g.f.: Sum_{n=-oo..+oo} x^n * exp(n*x) * (exp(n*x) - x^n)^n.

%C Compare e.g.f. to: Sum_{n=-oo..+oo} x^n * exp(n*x) * (1 - x^n*exp(n*x))^n = 0.

%H Paul D. Hanna, <a href="/A292807/b292807.txt">Table of n, a(n) for n = 1..100</a>

%F E.g.f.: Sum_{n=-oo..+oo} (-1)^n * x^(n^2-n) * exp((n^2-n)*x) / (exp(n*x) - x^n)^n.

%e E.g.f: A(x) = x + 7*x^2/2! + 37*x^3/3! + 807*x^4/4! + 13441*x^5/5! + 413243*x^6/6! + 13468813*x^7/7! + 571503103*x^8/8! + 28308826657*x^9/9! + 1666118229819*x^10/10! +...

%e Let E = exp(x), then A(x) = P(x) + Q(x) where

%e P(x) = 1 + (x*E)*(E - x) + (x*E)^2*(E^2 - x^2)^2 + (x*E)^3*(E^3 - x^3)^3 + (x*E)^4*(E^4 - x^4)^4 + (x*E)^5*(E^5 - x^5)^5 +...+ (x*E)^n*(E^n - x^n)^n +...

%e Q(x) = -1/(E - x) + (x*E)^2/(E^2 - x^2)^2 - (x*E)^6/(E^3 - x^3)^3 + (x*E)^12/(E^4 - x^4)^4 - (x*E)^20/(E^5 - x^5)^5 +...+ (-1)^n*(x*E)^(n^2-n)/(E^n - x^n)^n +...

%e Explicitly,

%e P(x) = 1 + x + 4*x^2/2! + 48*x^3/3! + 716*x^4/4! + 14580*x^5/5! + 399762*x^6/6! + 13652758*x^7/7! + 568482056*x^8/8! + 28365307128*x^9/9! + 1664953425350*x^10/10! +...

%e Q(x) = -1 + 3*x^2/2! - 11*x^3/3! + 91*x^4/4! - 1139*x^5/5! + 13481*x^6/6! - 183945*x^7/7! + 3021047*x^8/8! - 56480471*x^9/9! + 1164804469*x^10/10! +...

%o (PARI) {a(n) = my(A,P,Q, E=exp(x + x*O(x^n)));

%o P = sum(m=0,n,(x*E)^m*(E^m - x^m)^m);

%o Q = sum(m=1,n,(-1)^m*(x*E)^(m^2-m)/(E^m - x^m)^m);

%o A = P + Q; n!*polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A292088.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 04 2017