A217667 G.f.: Sum_{n>=0} (x + x^(2*n))^n.

1, 1, 2, 1, 1, 3, 1, 1, 5, 1, 1, 5, 1, 4, 6, 1, 1, 7, 8, 1, 8, 1, 1, 19, 1, 5, 10, 1, 16, 11, 1, 1, 23, 22, 1, 13, 1, 1, 42, 21, 1, 20, 1, 37, 16, 1, 36, 17, 46, 1, 34, 1, 1, 130, 1, 1, 20, 1, 67, 56, 85, 7, 22, 79, 1, 23, 1, 121, 185, 1, 1, 25, 23, 106, 191, 1, 1

Offset: 0

Author

Paul D. Hanna, Oct 10 2012

Keywords

• nonn

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.

Examples

			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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A217668, A217669.

Programs

Mathematica

terms = 100; Sum[(x + x^(2*n))^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)

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),", "))

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