A022573 Expansion of Product_{m>=1} (1+x^m)^8.
1, 8, 36, 128, 394, 1088, 2776, 6656, 15155, 33056, 69508, 141568, 280382, 541696, 1023512, 1895424, 3446617, 6163536, 10854400, 18846592, 32296742, 54673920, 91506000, 151523840, 248403014, 403396288, 649286724, 1036287744, 1640796160, 2578305024, 4022351720, 6232177664, 9592906446
Offset: 0
Keywords
Examples
G.f. = 1 + 8*x + 36*x^2 + 128*x^3 + 394*x^4 + 1088*x^5 + 2776*x^6 + ... G.f. = q + 8*q^4 + 36*q^7 + 128*q^10 + 394*q^13 + 1088*q^16 + 2776*q^19 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
Programs
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Magma
Coefficients(&*[(1+x^m)^8:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^8, {x, 0, n}]; (* Michael Somos, Feb 22 2015 *) With[{nmax=50}, CoefficientList[Series[Product[(1+q^k)^8, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 26 2018 *) -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^8, n))}; /* Michael Somos, May 29 2004 */ -
PARI
m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^8)) \\ G. C. Greubel, Feb 26 2018
Formula
G.f.: Product_{k>0} (1 + x^k)^8.
Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = -u^2 + v + 16*u*v^2. - Michael Somos, May 29 2004
Expansion of q^(-1/3) * (eta(q^2) / eta(q))^8 in powers of q.
Euler transform of period 2 sequence [ 8, 0, ...]. - Michael Somos, Jun 06 2005
Expansion of chi(-x)^-8 in powers of x where chi() is a Ramanujan theta function.
a(n) ~ exp(2 * Pi * sqrt(2*n/3)) / (16 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (8/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(8*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018