A036015 Number of partitions of n into parts not of form 4k+2, 8k, 8k+1 or 8k-1.
1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 6, 7, 7, 8, 10, 12, 13, 14, 17, 21, 22, 24, 29, 33, 36, 40, 46, 53, 58, 63, 73, 83, 90, 99, 113, 127, 138, 152, 171, 191, 209, 228, 255, 285, 309, 338, 377, 416, 453, 495, 547, 603, 656, 714, 787, 865, 938, 1020, 1121, 1226
Offset: 0
Examples
1 + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 4*x^12 + ...
References
- G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- S.-D. Chen and S.-S. Huang, On the series expansion of the Göllnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
- Nicolas Allen Smoot, A Partition Function Connected with the Göllnitz--Gordon Identities, arXiv:2005.09263 [math.NT], 2020. See g3(n) Table 2 p. 22.
- Eric Weisstein's World of Mathematics, Goellnitz-Gordon Identities
Programs
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Maple
M:=100; qf:=(a,q)->mul(1-a*q^j,j=0..M); tS:=1/(qf(q^3,q^8)*qf(q^4,q^8)*qf(q^5,q^8)); series(%,q,M); seriestolist(%);
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Mathematica
a[ n_] := SeriesCoefficient[ 1/ (QPochhammer[ q^3, q^8] QPochhammer[ q^4, q^8] QPochhammer[ q^5, q^8]), {q, 0, n}] (* Michael Somos, Jun 22 2012 *) a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[q^(k^2 + 2 k) QPochhammer[ -q, q^2, k] / QPochhammer[ q^2, q^2, k], {k, 0, Sqrt[n + 1] - 1}], {q, 0, n}]] (* Michael Somos, Jun 22 2012 *) nmax=60; CoefficientList[Series[Product[1/((1-x^(8*k-3))*(1-x^(8*k-4))*(1-x^(8*k-5))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2015 *) -
PARI
{a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - ([ 0, 0, 1, 1, 1, 0, 0, 0][(k-1)%8 + 1]) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jun 28 2004 */
Formula
Let qf(a, q) = Product(1-a*q^j, j=0..infinity); g.f. is 1/(qf(q^3, q^8)*qf(q^4, q^8)*qf(q^5, q^8)).
a(n) ~ sqrt(2-sqrt(2)) * exp(sqrt(n)*Pi/2) / (8*n^(3/4)). - Vaclav Kotesovec, Oct 04 2015
Extensions
a(64) corrected by Vaclav Kotesovec, Oct 04 2015
Comments