OFFSET
0,3
COMMENTS
More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),
for any fixed integer k; here, k = 1 and y = 1, F = x, G = x^2.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
FORMULA
Generating functions.
(1) Sum_{n>=0} (x + x^(2*n))^n.
(2) Sum_{n>=0} x^(n^2) / (1 - x^(2*n+1))^(n+1). - Paul D. Hanna, Jun 02 2019
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 + x^7 + 5*x^8 + x^9 +...
where
A(x) = 1 + (x + x^2) + (x + x^4)^2 + (x + x^6)^3 + (x + x^8)^4 + (x + x^10)^5 +...
Also
A(x) = 1/(1-x) + x/(1 - x^3)^2 + x^4/(1 - x^5)^3 + x^9/(1 - x^7)^4 + x^16/(1 - x^9)^5 + x^25/(1 - x^11)^6 + x^36/(1 - x^13)^7 + x^49/(1 - x^15)^8 + ...
MATHEMATICA
terms = 100; Sum[(x + x^(2*n))^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, (x+x^(2*m) +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 10 2012
STATUS
approved