A357539 a(n) = coefficient of x^n/n! in: Sum_{n>=0} ( x*exp(x) )^(n*(n+1)/2).

1, 1, 2, 9, 76, 545, 3966, 47257, 807416, 13431105, 201158650, 2992272041, 55015365252, 1383804654817, 39956273419622, 1127353750507545, 29721911064179056, 748976662158153857, 19509333366569811570, 592071561505183956553, 22102320673776378606140

Offset

0,3

Comments

Conjecture: Limit_{n->infinity} (a(n)/n!)^(1/n) = 1/LambertW(1). - Vaclav Kotesovec, Dec 06 2022

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Vaclav Kotesovec, Plot of a(n+1)/a(n)/n for n = 1..10000

Vaclav Kotesovec, Plot of a(n) / (n^n/(exp(n)*LambertW(1)^n)) for n = 1..10000

Formula

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

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

(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.

Example

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! + ...
where
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) + ...
The e.g.f. also equals the infinite product:
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)) * ...

Mathematica

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 *)

Prog

(PARI) {a(n) = my(A=1);
A = sum(m=0, sqrtint(2*n+9), (x * exp(x +x*O(x^n)))^(m*(m+1)/2) ); n! * polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A370259 A335378 A232471 * A277181 A105785 A245406

Adjacent sequences: A357536 A357537 A357538 * A357540 A357541 A357542

Keyword

nonn

Author

Paul D. Hanna, Dec 05 2022

Status

approved