A160571 G.f.: Product_{n>=1} (1 + x^n + x^(n+1)).
1, 1, 2, 3, 5, 7, 10, 15, 21, 28, 38, 52, 70, 92, 119, 154, 200, 258, 329, 416, 523, 655, 819, 1022, 1269, 1566, 1924, 2357, 2879, 3507, 4263, 5170, 6250, 7530, 9048, 10849, 12980, 15496, 18466, 21967, 26079, 30894, 36526, 43109, 50792, 59743, 70160
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 10*x^6 + 15*x^7 + ... G.f.: A(x) = (1+x*(1+x))*(1+x^2*(1+x))*(1+x^3*(1+x))*(1+x^4*(1+x))*... G.f.: A(x) = (1+x*(1+x)) + x^2*(1+x)*(1 + x^3*(1+x))*(1+x*(1+x))/(1-x) + x^7*(1+x)^2*(1 + x^5*(1+x))*(1+x*(1+x))*(1+x^2*(1+x))/((1-x)*(1-x^2)) + x^15*(1+x)^3*(1 + x^7*(1+x))*(1+x*(1+x))*(1+x^2*(1+x))*(1+x^3*(1+x))/((1-x)*(1-x^2)*(1-x^3)) + ... G.f.: A(x) = 1 + x*(1+x)/(1-x) + x^3*(1+x)^2/((1-x)*(1-x^2)) + x^6*(1+x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*(1+x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + ...
Links
- Robert Israel, Table of n, a(n) for n = 0..10000
- MathOverflow, Being even or odd in the product expansion, 2017.
Programs
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Maple
N:= 100: # for a(0)..a(N) P:= mul(1+x^n+x^(n+1),n=1..N): S:= series(P,x,N+1): seq(coeff(S,x,j),j=0..N); # Robert Israel, Sep 04 2018
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Mathematica
With[{nn=50},CoefficientList[Series[Product[1+x^n+x^(n+1),{n,1,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Dec 29 2015 *) -
PARI
a(n)=polcoeff(prod(k=1,n,1+x^k*(1+x) +x*O(x^n)),n)
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PARI
{a(n)=local(A=1+x); A=sum(m=0, n, x^(m*(3*m+1)/2)*(1+x)^m*(1 + x^(2*m+1)*A)*prod(k=1, m, (1+A*x^k)/(1-x^k+x*O(x^n)))); polcoeff(A, n)} -
PARI
{a(n)=local(A=1+x); A=sum(m=0, n, x^(m*(m+1)/2)*(1+x)^m/prod(k=1, m, 1-x^k +x*O(x^n))); polcoeff(A, n)}
Formula
G.f.: A(x) = Sum_{n>=0} x^(n*(3*n+1)/2)*(1+x)^n*(1 + x^(2*n+1)*(1+x)) * Product_{k=1..n} (1 + x^k*(1+x))/(1-x^k) due to Sylvester's identity.
G.f.: A(x) = Sum_{n>=0} x^(n*(n+1)/2)*(1+x)^n / Product_{k=1..n} (1-x^k).
G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)*(1 + x)^d/d). - Ilya Gutkovskiy, Apr 18 2019
a(n) ~ c * exp(r*sqrt(n)) / n^(3/4), where r = 2*sqrt(-polylog(2,-2)) = 2.397287105779... and c = (-polylog(2,-2))^(1/4) / (6*sqrt(Pi)) = 0.10294821957... - Vaclav Kotesovec, Oct 24 2020, updated Jun 25 2021