A338176 G.f.: Sum_{n>=0} Product_{k=0..n-1} ( (1+x)^(n+k) - 1 ).

1, 1, 6, 69, 1115, 23271, 594755, 17983607, 627836008, 24851815494, 1099770494540, 53802588161113, 2883239412214382, 167966313901092777, 10568851970059809742, 714331456391338800032, 51613269668720039244727, 3970054568747254273095950, 323894316366314518137627174, 27935642142397030897774502220

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f.: Sum_{n>=0} Product_{k=0..n-1} ( (1+x)^(n+k) - 1 ).

G.f.: Sum_{n>=0} (1+x)^(n*(3*n-1)/2) / Product_{k=0..n} (1 + (1+x)^(n+k)).

a(n) ~ c * d^n * n! / sqrt(n), where d = 4.66344230824842673353729480791793... and c = 0.197562786708339663216376083766... - Vaclav Kotesovec, Oct 15 2020

Example

G.f.: A(x) = 1 + x + 6*x^2 + 69*x^3 + 1115*x^4 + 23271*x^5 + 594755*x^6 + 17983607*x^7 + 627836008*x^8 + 24851815494*x^9 + 1099770494540*x^10 + ...
where
A(x) = 1 + ((1+x) - 1) + ((1+x)^2 - 1)*((1+x)^3 - 1) + ((1+x)^3 - 1)*((1+x)^4 - 1)*((1+x)^5 - 1) + ((1+x)^4 - 1)*((1+x)^5 - 1)*((1+x)^6 - 1)*((1+x)^7 - 1) + ((1+x)^5 - 1)*((1+x)^6 - 1)*((1+x)^7 - 1)*((1+x)^8 - 1)*((1+x)^9 - 1) + ... + Product_{k=0..n-1} ((1+x)^(n+k) - 1) + ...
Also
A(x) = 1/(1 + 1)  +  (1+x)/((1 + (1+x))*(1 + (1+x)^2))  +  (1+x)^5/((1 + (1+x)^2)*(1 + (1+x)^3)*(1 + (1+x)^4))  +  (1+x)^12/((1 + (1+x)^3)*(1 + (1+x)^4)*(1 + (1+x)^5)*(1 + (1+x)^6))  +  (1+x)^22/((1 + (1+x)^4)*(1 + (1+x)^5)*(1 + (1+x)^6)*(1 + (1+x)^7)*(1 + (1+x)^8)) + ... + (1+x)^(n*(3*n-1)/2)/( Product_{k=0..n} (1 + (1+x)^(n+k)) ) + ...

Mathematica

nmax = 20; CoefficientList[Series[Sum[Product[(1+x)^(n+k) - 1, {k, 0, n-1}], {n, 0, nmax-1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 15 2020 *)

Prog

(PARI) {a(n) = polcoeff( sum(m=0, n, prod(k=0, m-1, (1+x)^(m+k) - 1 +x*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A347674 A234509 A177751 * A235327 A349556 A364982

Adjacent sequences: A338173 A338174 A338175 * A338177 A338178 A338179

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2020

Status

approved