OFFSET
0,3
COMMENTS
Equals the row sums of triangle A214398.
a(n) is the number of weak compositions of n such that if the first part is equal to k then there are a total of k^2 + 1 parts. A weak composition is an ordered partition of the integer n into nonnegative parts. a(3) = 6 because we have: 1+2, 2+0+0+0+1, 2+0+0+1+0, 2+0+1+0+0, 2+1+0+0+0, 3+0+0+0+0+0+0+0+0+0. - Geoffrey Critzer, Oct 09 2013
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..250
FORMULA
a(n) = Sum_{k=0..n} C((n-k)^2 + k-1, k).
G.f.: A(x) = Sum_{n>=0} (x-x^2)^n*Product_{k=1..n} ((1-x)^(4*k-3) - x)/((1-x)^(4*k-1) - x) due to a q-series identity.
Let q = 1/(1-x), then g.f. A(x) equals the continued fraction:
. A(x) = 1/(1- q*x/(1- q*(q^2-1)*x/(1- q^5*x/(1- q^3*(q^4-1)*x/(1- q^9*x/(1- q^5*(q^6-1)*x/(1- q^13*x/(1- q^7*(q^8-1)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
log(a(n)) ~ n*(log(n) - 2) * (1 + log(4*n) - log((log(n) - 2)*log(n))) / log(n). - Vaclav Kotesovec, Jan 10 2023
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 83*x^5 + 363*x^6 +...
A(x) = 1 + (x-x^2)*((1-x)-x)/((1-x)^3-x) + (x-x^2)^2*((1-x)-x)*((1-x)^5-x)/(((1-x)^3-x)*((1-x)^7-x)) + (x-x^2)^3*((1-x)-x)*((1-x)^5-x)*((1-x)^9-x)/(((1-x)^3-x)*((1-x)^7-x)*((1-x)^11-x)) +...
MATHEMATICA
nn=22; CoefficientList[Series[Sum[x^k/(1-x)^(k^2), {k, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Oct 09 2013 *)
PROG
(PARI) {a(n)=sum(k=0, n, binomial((n-k)^2+k-1, k))}
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m/(1-x+x*O(x^n))^(m^2)), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, (x-x^2)^m*prod(k=1, m, ((1-x)^(4*k-3)-x)/((1-x)^(4*k-1)-x +x*O(x^n)))), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 21 2010
STATUS
approved