A217668 G.f.: Sum_{n>=0} x^n*(1 + x^n)^n.
1, 1, 2, 1, 3, 1, 5, 1, 5, 4, 6, 1, 14, 1, 8, 11, 13, 1, 25, 1, 22, 22, 12, 1, 61, 6, 14, 37, 50, 1, 77, 1, 73, 56, 18, 36, 175, 1, 20, 79, 211, 1, 135, 1, 188, 232, 24, 1, 421, 8, 236, 137, 313, 1, 307, 331, 422, 172, 30, 1, 1423, 1, 32, 295, 601, 716, 727, 1
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 +... where we have the following series identity: A(x) = 1 + x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5 + x^6*(1+x^6)^6 + x^7*(1+x^7)^7 + x^8*(1+x^8)^8 + x^9*(1+x^9)^9 +... A(x) = 1/(1-x) + x^2/(1-x^2)^2 + x^6/(1-x^3)^3 + x^12/(1-x^4)^4 + x^20/(1-x^5)^5 + x^30/(1-x^6)^6 + x^42/(1-x^7)^7 + x^56/(1-x^8)^8 +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
terms = 100; Sum[x^n*(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *) -
PARI
{a(n,t=1)=polcoeff(sum(m=0,n,x^m*(t+x^m +x*O(x^n))^m),n)} for(n=0,100,print1(a(n),", ")) -
PARI
{a(n,t=1)=local(A=1+x); A=sum(k=0, sqrtint(n+1), x^(k*(k+1))/(1 - t*x^(k+1) +x*O(x^n))^(k+1) ); polcoeff(A, n)} for(n=0,100,print1(a(n),", ")) \\ Paul D. Hanna, Sep 13 2014 -
PARI
{a(n) = if(n==0,1, sumdiv(n,d, binomial(n/d,d-1)) )} for(n=0,100,print1(a(n),", ")) \\ Paul D. Hanna, Apr 25 2018
Formula
G.f.: Sum_{n>=0} x^(n*(n+1)) / (1 - x^(n+1))^(n+1). - Paul D. Hanna, Sep 13 2014
a(n) = Sum_{d|n} binomial(n/d, d-1) for n>0 with a(0) = 1. - Paul D. Hanna, Apr 25 2018