A220352 G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n^2).

1, 1, 3, 16, 118, 1116, 12869, 175096, 2745726, 48756438, 967026762, 21188546616, 508286084222, 13249410224210, 372908807794347, 11270832179901016, 364083312029454453, 12518063823862065816, 456432182550333723335, 17591590487681007523476

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

G.f.: Sum_{n>=1} ((1+x)^n - 1)^(n-1) / (1+x)^(n^2).

a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 1.44832302735058524286860126583754380692... . - Vaclav Kotesovec, Nov 08 2014

In closed form, c = 1 / (log(2) * sqrt(1-log(2)) * 2^((1+log(2))/2)). - Vaclav Kotesovec, May 03 2015

Example

G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 118*x^4 + 1116*x^5 + 12869*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x/(1+x) + (2*x + x^2)^2/(1+x)^4 + (3*x + 3*x^2 + x^3)^3/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^4/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^5/(1+x)^25 +...
(2) A(x) = 1/(1+x) + (2*x + x^2)/(1+x)^4 + (3*x + 3*x^2 + x^3)^2/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^3/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^4/(1+x)^25 +...

Prog

(PARI) {a(n)=local(q=1+x+x*O(x^n), A=1); A=sum(k=0, n, q^(-k^2)*(q^k-1)^k); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(q=1+x+x*O(x^n), A=1); A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1)); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A220353, A121886, A187826.

Sequence in context: A074522 A302701 A190633 * A333682 A125807 A125222

Adjacent sequences: A220349 A220350 A220351 * A220353 A220354 A220355

Keyword

nonn

Author

Paul D. Hanna, Dec 11 2012

Status

approved