OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..100
FORMULA
E.g.f.: Sum_{n>=0} Product_{k=0..n-1} ( exp((n+k)*x) - 1 ).
E.g.f.: Sum_{n>=0} exp(n*(3*n-1)/2 * x) / Product_{k=0..n} (1 + exp((n+k)*x)).
a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 4.66344230824842673353729480791793... and c = 0.2199220967955858347917680132696... - Vaclav Kotesovec, Oct 15 2020
EXAMPLE
E.g.f.: A(x) = 1 + x + 13*x^2/2! + 451*x^3/3! + 29329*x^4/4! + 3070651*x^5/5! + 471888433*x^6/6! + 100030519051*x^7/7! + 27969080006209*x^8/8! + ...
where
A(x) = 1 + (exp(x) - 1) + (exp(x)^2 - 1)*(exp(3*x) - 1) + (exp(3*x) - 1)*(exp(4*x) - 1)*(exp(5*x) - 1) + (exp(4*x) - 1)*(exp(5*x) - 1)*(exp(6*x) - 1)*(exp(7*x) - 1) + (exp(5*x) - 1)*(exp(6*x) - 1)*(exp(7*x) - 1)*(exp(8*x) - 1)*(exp(9*x) - 1) + ... + Product_{k=0..n-1} (exp((n+k)*x) - 1) + ...
Also
A(x) = 1/(1 + 1) + exp(x)/((1 + exp(x))*(1 + exp(2*x))) + exp(5*x)/((1 + exp(2*x))*(1 + exp(3*x))*(1 + exp(4*x))) + exp(12*x)/((1 + exp(3*x))*(1 + exp(4*x))*(1 + exp(5*x))*(1 + exp(6*x))) + exp(22*x)/((1 + exp(4*x))*(1 + exp(5*x))*(1 + exp(6*x))*(1 + exp(7*x))*(1 + exp(8*x))) + ... + exp((n*(3*n-1)/2*x)/( Product_{k=0..n} (1 + exp((n+k)*x)) ) + ...
MATHEMATICA
nmax = 20; CoefficientList[Series[Sum[Product[E^((n+k)*x) - 1, {k, 0, n-1}], {n, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 15 2020 *)
PROG
(PARI) {a(n) = n!*polcoeff( sum(m=0, n, prod(k=0, m-1, exp((m+k)*x +x*O(x^n)) - 1) ), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2020
STATUS
approved