A217669 - OEIS

%I #16 Jun 02 2019 21:39:27

%S 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,

%T 72,1,113,1,102,67,19,72,239,1,21,92,313,1,191,1,244,331,25,1,575,29,

%U 357,154,392,1,452,496,577,191,31,1,1979,1,33,443,750,1002

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

%C More generally, the following sums are equal:

%C (1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,

%C (2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),

%C for any fixed integer k; here, k = 1 and y = 1, F = x, G = x.

%H Paul D. Hanna, <a href="/A217669/b217669.txt">Table of n, a(n) for n = 0..1000</a>

%F Generating functions.

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

%F (2) Sum_{n>=0} x^(n^2) / (1 - x^(n+1))^(n+1). - _Paul D. Hanna_, Jun 02 2019

%e 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 +...

%e where

%e A(x) = 1 + (x + x) + (x + x^2)^2 + (x + x^3)^3 + (x + x^4)^4 + (x + x^5)^5 +...

%e Also

%e 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 + ...

%o (PARI) {a(n)=polcoeff(sum(m=0,n,(x+x^m +x*O(x^n))^m),n)}

%o for(n=0,100,print1(a(n),", "))

%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+1)), n)}

%o for(n=0, 100, print1(a(n), ", "))

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 10 2012