A022569 Expansion of Product_{m>=1} (1+x^m)^4.
1, 4, 10, 24, 51, 100, 190, 344, 601, 1024, 1702, 2768, 4422, 6948, 10752, 16424, 24782, 36972, 54602, 79872, 115805, 166540, 237664, 336720, 473856, 662596, 920934, 1272728, 1749407, 2392268, 3255410, 4409344, 5945730, 7983388, 10675712, 14220240, 18870672, 24951740, 32878114
Offset: 0
Keywords
Examples
G.f. = 1 + 4*x + 10*x^2 + 24*x^3 + 51*x^4 + 100*x^5 + 190*x^6 + 344*x^7 + ... G.f. = q + 4*q^7 + 10*q^13 + 24*q^19 + 51*q^25 + 100*q^31 + 190*q^37 + 344*q^43 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
-
Magma
Coefficients(&*[(1+x^m)^4:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
-
Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-4, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *) a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^4, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^4, n))}; /* Michael Somos, Apr 26 2008 */ -
PARI
m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^4)) \\ G. C. Greubel, Feb 26 2018
Formula
Expansion of q^(-1/6) * (eta(q^2) / eta(q))^4 in powers of q.
Expansion of chi(-q)^(-4) in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 2 sequence [ 4, 0, ...]. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(q) = (A(q^6) * q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v * (1 + 16 * u * v) - u^2. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(x) = A(q^6) * q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v * (u^2 - v) - 4 * w^2 * (u^2 + v). - Michael Somos, Apr 26 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022599.
G.f.: Product_{k>0} (1 + x^k)^4.
Convolution inverse of A022599.
G.f.: T(0)/x, where T(k) = 1 - 1/(1 - (1+(x)^(k+1))^4/((1+(x)^(k+1))^4 - 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 07 2013
a(n) ~ exp(2 * Pi * sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)) * (1 + (Pi/(6*sqrt(3)) - 3*sqrt(3)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Mar 05 2015, extended Jan 16 2017
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(4*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
Comments