A357539 - OEIS

%I #21 Dec 06 2022 07:14:17

%S 1,1,2,9,76,545,3966,47257,807416,13431105,201158650,2992272041,

%T 55015365252,1383804654817,39956273419622,1127353750507545,

%U 29721911064179056,748976662158153857,19509333366569811570,592071561505183956553,22102320673776378606140

%N a(n) = coefficient of x^n/n! in: Sum_{n>=0} ( x*exp(x) )^(n*(n+1)/2).

%C Conjecture: Limit_{n->infinity} (a(n)/n!)^(1/n) = 1/LambertW(1). - _Vaclav Kotesovec_, Dec 06 2022

%H Paul D. Hanna, <a href="/A357539/b357539.txt">Table of n, a(n) for n = 0..500</a>

%H Vaclav Kotesovec, <a href="/A357539/a357539_1.jpg">Plot of a(n+1)/a(n)/n for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A357539/a357539_2.jpg">Plot of a(n) / (n^n/(exp(n)*LambertW(1)^n)) for n = 1..10000</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined by the following expressions.

%F (1) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * exp(n*(n+1)/2 * x).

%F (2) A(x) = Product_{n>=1} (1 + x^n*exp(n*x)) * (1 - x^(2*n)*exp(2*n*x)), by the Jacobi triple product identity.

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 545*x^5/5! + 3966*x^6/6! + 47257*x^7/7! + 807416*x^8/8! + 13431105*x^9/9! + 201158650*x^10/10! + ...

%e where

%e A(x) = 1 + (x*exp(x)) + (x*exp(x))^3 + (x*exp(x))^6 + (x*exp(x))^10 + (x*exp(x))^15 + (x*exp(x))^21 + ... + (x*exp(x))^(n*(n+1)/2) + ...

%e The e.g.f. also equals the infinite product:

%e A(x) = (1 + x*exp(x))*(1 - x^2*exp(2*x)) * (1 + x^2*exp(2*x))*(1 - x^4*exp(4*x)) * (1 + x^3*exp(3*x))*(1 - x^6*exp(6*x)) * (1 + x^4*exp(4*x))*(1 - x^8*exp(8*x)) * ... * (1 + x^n*exp(n*x))*(1 - x^(2*n)*exp(2*n*x)) * ...

%t nmax = 20; CoefficientList[Series[Sum[(x*E^x)^(k*(k + 1)/2), {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Dec 06 2022 *)

%o (PARI) {a(n) = my(A=1);

%o A = sum(m=0,sqrtint(2*n+9), (x * exp(x +x*O(x^n)))^(m*(m+1)/2) ); n! * polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 05 2022