A217669 G.f.: Sum_{n>=0} (x + x^n)^n.

1, 2, 1, 3, 2, 4, 1, 8, 1, 7, 7, 7, 1, 22, 1, 9, 17, 20, 1, 32, 1, 37, 29, 13, 1, 86, 16, 15, 46, 72, 1, 113, 1, 102, 67, 19, 72, 239, 1, 21, 92, 313, 1, 191, 1, 244, 331, 25, 1, 575, 29, 357, 154, 392, 1, 452, 496, 577, 191, 31, 1, 1979, 1, 33, 443, 750, 1002

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.

Examples

			G.f.: A(x) = 1 + 2*x + x^2 + 3*x^3 + 2*x^4 + 4*x^5 + x^6 + 8*x^7 + x^8 +...
where
A(x) = 1 + (x + x) + (x + x^2)^2 + (x + x^3)^3 + (x + x^4)^4 + (x + x^5)^5 +...
Also
A(x) = 1/(1-x) + x/(1 - x^2)^2 + x^4/(1 - x^3)^3 + x^9/(1 - x^4)^4 + x^16/(1 - x^5)^5 + x^25/(1 - x^6)^6 + x^36/(1 - x^7)^7 + x^49/(1 - x^8)^8 + ...
		

Links

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

Crossrefs

Cf. A217667, A217668, A325997, A325998, A325999.

Programs

PARI

{a(n)=polcoeff(sum(m=0,n,(x+x^m +x*O(x^n))^m),n)}
for(n=0,100,print1(a(n),", "))

PARI

{a(n)=polcoeff(sum(m=0, n, x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+1)), n)}
for(n=0, 100, print1(a(n), ", "))

Formula

Generating functions.

(1) Sum_{n>=0} (x + x^n)^n.

(2) Sum_{n>=0} x^(n^2) / (1 - x^(n+1))^(n+1). - Paul D. Hanna, Jun 02 2019