cp's OEIS Frontend

A036015 Number of partitions of n into parts not of form 4k+2, 8k, 8k+1 or 8k-1.

%I A036015 #34 Feb 16 2025 08:32:37
%S A036015 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,
%T A036015 36,40,46,53,58,63,73,83,90,99,113,127,138,152,171,191,209,228,255,
%U A036015 285,309,338,377,416,453,495,547,603,656,714,787,865,938,1020,1121,1226
%N A036015 Number of partitions of n into parts not of form 4k+2, 8k, 8k+1 or 8k-1.
%C A036015 Case k=2,i=1 of Gordon/Goellnitz/Andrews Theorem.
%C A036015 Also number of partitions in which no odd part is repeated, with no part of size less than or equal to 2 and where differences between adjacent parts are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.
%C A036015 Euler transform of period 8 sequence [ 0, 0, 1, 1, 1, 0, 0, 0, ...]. - _Michael Somos_, Jun 28 2004
%D A036015 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
%H A036015 Seiichi Manyama, <a href="/A036015/b036015.txt">Table of n, a(n) for n = 0..10000</a>
%H A036015 S.-D. Chen and S.-S. Huang, <a href="https://doi.org/10.1142/S1793042105000030">On the series expansion of the Göllnitz-Gordon continued fraction</a>, Internat. J. Number Theory, 1 (2005), 53-63.
%H A036015 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%H A036015 Nicolas Allen Smoot, <a href="https://arxiv.org/abs/2005.09263">A Partition Function Connected with the Göllnitz--Gordon Identities</a>, arXiv:2005.09263 [math.NT], 2020. See g3(n) Table 2 p. 22.
%H A036015 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Goellnitz-GordonIdentities.html">Goellnitz-Gordon Identities</a>
%F A036015 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)).
%F A036015 a(n) ~ sqrt(2-sqrt(2)) * exp(sqrt(n)*Pi/2) / (8*n^(3/4)). - _Vaclav Kotesovec_, Oct 04 2015
%e A036015 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 + ...
%p A036015 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(%);
%t A036015 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 *)
%t A036015 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 *)
%t A036015 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 *)
%o A036015 (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 */
%Y A036015 Cf. A036016, A316384.
%K A036015 nonn,easy
%O A036015 0,9
%A A036015 _Olivier Gérard_
%E A036015 a(64) corrected by _Vaclav Kotesovec_, Oct 04 2015