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A069910 Expansion of Product_{i in A069908} 1/(1 - x^i).

%I A069910 #71 Feb 16 2025 08:32:46
%S A069910 1,0,1,1,2,2,3,3,5,5,7,8,11,12,16,18,23,26,33,37,46,52,63,72,87,98,
%T A069910 117,133,157,178,209,236,276,312,361,408,471,530,609,686,784,881,1004,
%U A069910 1126,1279,1433,1621,1814,2048,2286,2574,2871,3223,3590,4022,4472,5000
%N A069910 Expansion of Product_{i in A069908} 1/(1 - x^i).
%C A069910 Number 39 of the 130 identities listed in Slater 1952.
%C A069910 Number of partitions of 2*n into distinct odd parts. - _Vladeta Jovovic_, May 08 2003
%D A069910 M. D. Hirschhorn, The Power of q, Springer, 2017. Chapter 19, Exercises p. 173.
%H A069910 Alois P. Heinz, <a href="/A069910/b069910.txt">Table of n, a(n) for n = 0..1000</a>
%H A069910 G. E. Andrews et al., <a href="http://dx.doi.org/10.1080/16073606.2001.9639228">q-Engel series expansions and Slater's identities</a>, Quaestiones Math., 24 (2001), 403-416.
%H A069910 George E. Andrews, Jethro van Ekeren and Reimundo Heluani, <a href="https://arxiv.org/abs/2005.10769">The singular support of the Ising model</a>, arXiv:2005.10769 [math.QA], 2020. See (1.4.2) p. 2.
%H A069910 T. Gannon, G. Hoehn, H. Yamauchi, et. al., <a href="https://www.math.ksu.edu/~gerald/voas/voa/minimal1.html">VOA Unitary Minimal Model m=1</a>, character.
%H A069910 M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/0097-3165(79)90005-0">Some partition theorems of the Rogers-Ramanujan type</a>, J. Combin. Theory Ser. A 27 (1979), no. 1, 33-37. MR0541341 (80j:05010). See Theorem 4. [From _N. J. A. Sloane_, Mar 19 2012]
%H A069910 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 A069910 Lucy Joan Slater, <a href="https://doi.org/10.1112/plms/s2-54.2.147">Further Identities of the Rogers-Ramanujan Type</a>, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp. 147-167, (1952).
%H A069910 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Jackson-SlaterIdentity.html">Jackson-Slater Identity</a>
%H A069910 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A069910 M. P. Zaletel and R. S. K. Mong, <a href="http://arxiv.org/abs/1208.4862">Exact Matrix Product States for Quantum Hall Wave Functions</a>, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - _N. J. A. Sloane_, Dec 25 2012
%F A069910 Euler transform of period 16 sequence [0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, ...]. - _Michael Somos_, Apr 11 2004
%F A069910 G.f.: Sum_{n>=0} q^(2*n^2) / Product_{k=1..2*n} (1 - q^k). - _Joerg Arndt_, Apr 01 2014
%F A069910 a(n) ~ exp(sqrt(n/3)*Pi) / (2^(5/2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 04 2015
%F A069910 Expansion of f(x^3, x^5) / f(-x^2) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Apr 14 2016
%F A069910 a(n) = A000700(2*n).
%F A069910 a(n) = A027356(4n+1,2n+1). - _Alois P. Heinz_, Oct 28 2019
%F A069910 From _Peter Bala_, Feb 08 2021: (Start)
%F A069910 G.f.: A(x) = Product_{n >= 1} (1 + x^(4*n))^2*(1 + x^(4*n-2))*(1 + x^(8*n-3))*(1 + x^(8*n-5)).
%F A069910 The 2 X 2 matrix Product_{k >= 0} [1, x^(2*k+1); x^(2*k+1), 1] = [A(x^2), x*B(x^2); x*B(x)^2, A(x^2)], where B(x) is the g.f. of A069911.
%F A069910 A(x^2) + x*B(x^2) = A^2(-x) + x*B^2(-x) = Product_{k >= 0} 1 + x^(2*k+1), the g.f. of A000700.
%F A069910 A^2(x) + x*B^2(x) is the g.f. of A226622.
%F A069910 (A^2(x) + x*B^2(x))/(A^2(x) - x*B^2(x)) is the g.f. of A208850.
%F A069910 A^4(sqrt(x)) - x*B^4(sqrt(x)) is the g.f. of A029552.
%F A069910 A(x)*B(x) is the g.f. of A226635; A(-x)/B(-x) is the g.f. of A111374; B(-x)/A(-x) is the g.f. of A092869. (End)
%e A069910 G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + ...
%e A069910 G.f. = q^-1 + q^95 + q^143 + 2*q^191 + 2*q^239 + 3*q^287 + 3*q^335 + ...
%p A069910 a:= proc(n) option remember; `if`(n=0, 1,
%p A069910       add(add(d*[0$2, 1$4, 0$5, 1$4, 0][irem(d, 16)+1],
%p A069910       d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A069910     end:
%p A069910 seq(a(n), n=0..80);  # _Alois P. Heinz_, Apr 01 2014
%t A069910 max = 56; p = Product[1/(1-x^i), {i, Select[Range[max], MemberQ[{2, 3, 4, 5, 11, 12, 13, 14}, Mod[#, 16]]&]}]; s = Series[p, {x, 0, max}]; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* _Jean-François Alcover_, Apr 09 2014 *)
%t A069910 nmax=60; CoefficientList[Series[Product[(1-x^(8*k-1))*(1-x^(8*k-7))*(1-x^(8*k))*(1-x^(16*k-6))*(1-x^(16*k-10))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 04 2015 *)
%t A069910 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0 }[[ Mod[k, 16] + 1]], {k, n}], {x, 0, n}]; (* _Michael Somos_, Apr 14 2016 *)
%o A069910 (PARI) {a(n) = my(A); if( n<0,0, n=2*n; A = x * O(x^n); polcoeff( eta(-x + A) / eta(x^2 + A), n))}; /* _Michael Somos_, Apr 11 2004 */
%o A069910 (PARI) N=66;  q='q+O('q^N);  S=1+sqrtint(N);
%o A069910 gf=sum(n=0, S, q^(2*n^2) / prod(k=1, 2*n, 1-q^k ) );
%o A069910 Vec(gf)  \\ _Joerg Arndt_, Apr 01 2014
%o A069910 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0][k%16 + 1]), n))}; /* _Michael Somos_, Apr 14 2016 */
%Y A069910 Cf. A000700, A027356, A035457, A069908, A069909, A069911, A122129, A122135.
%K A069910 nonn,easy
%O A069910 0,5
%A A069910 _N. J. A. Sloane_, May 05 2002