%I #25 Nov 12 2017 03:58:18

%S 1,0,1,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19,23,26,31,35,42,47,55,62,

%T 72,81,94,105,121,136,155,175,199,222,252,282,318,355,400,445,501,556,

%U 624,693,774,857,957,1059,1178,1302,1446,1596,1769,1951,2158,2376,2624,2885,3182,3495,3847,4221,4642

%N Expansion of H(q)*G(q^11) where H and G are respectively the g.f. of A003106 and A003114 (Rogers-Ramanujan functions).

%C Also the expansion of 1+q^2*H(q^11)*G(q); that is, H(q)*G(q^11) - q^2*G(q)*H(q^11) = 1, we also have H(q)*G(q)^11 - q^2*G(q)*H(q)^11 = 1 + 11*q*(G(q)*H(q))^6, see the Ramanujan reference.

%C Number of partitions of n into parts t such that t mod 55 is in {2, 3, 7, 8, 11, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 47, 48, 52, 53}.

%C With E(q) = Product_{n>=1} (1-q^n) we have G(q)*H(q) - E(q^5)/E(q), G(q) = ( E(q^8)/E(q^2) * (G(q^16) + q*H(-q^4)) ), and H(q) = ( E(q^8)/E(q^2) * (q^3*H(q^16) + G(-q^4)) ), see the Berkovich/Yesilyurt reference.

%H Alexander Berkovich, Hamza Yesilyurt, <a href="http://arxiv.org/abs/1204.1092">On Rogers-Ramanujan functions, binary quadratic forms and eta-quotients</a>, arXiv:1204.1092v2 [math.NT], 2012.

%H Srinivasa Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper29/page1.htm">Algebraic relations between certain infinite products</a>, Proceedings of the London Mathematical Society, vol.2, no.18, 1920.

%F G.f.: H(q)*G(q^11) where G(q) = Sum_{n>=0} q^(n^2)/Product_{k=1..n} (1-q^k) and H(q) = Sum_{n>=0} q^(n^2+n)/Product_{k=1..n} (1-q^k).

%F G.f.: 1 / Product_{k>=0} (1 - q^k) where k (mod 55) is restricted to the set {2, 3, 7, 8, 11, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 47, 48, 52, 53} (the set has 24 elements).

%o (PARI)

%o N=66; q='q+O('q^N );

%o S=2+2*ceil(sqrt(N));

%o G(q)=sum(n=0,S,q^(n^2)/prod(k=1,n,1-q^k)); /* g.f. of A003114 */

%o H(q)=sum(n=0,S,q^(n^2+n)/prod(k=1,n,1-q^k)); /* g.f. of A003106 */

%o Vec(H(q)*G(q^11)) /* show terms */

%o /* checking the modular equations, all expressions are zero:

%o ( H(q)*G(q)^11 - q^2*G(q)*H(q)^11 ) - ( 1 + 11*q*(G(q)*H(q))^6 )

%o ( H(q)*G(q^11) - q^2*G(q)*H(q^11) ) - ( 1 )

%o E(q)=prod(n=1,N, 1-q^n);

%o G(q)*H(q) - E(q^5)/E(q)

%o G(q) - ( E(q^8)/E(q^2) * (G(q^16) + q*H(-q^4)) )

%o H(q) - ( E(q^8)/E(q^2) * (q^3*H(q^16) + G(-q^4)) )

%o */

%o (PARI)

%o N=66; q='q+O('q^N );

%o E=[2,3,7,8,11,12,13,17,18,22,23,27,28,32,33,37,38,42,43,44,47,48,52,53];

%o Vec( 1 / prod(K=0, N\55+1, prod(k=1,24, 1 - q^(K*55+E[k]) ) ) )

%Y Cf. A003106 and A003114 (Rogers-Ramanujan functions H and G).

%Y Cf. A121591 (expansion of q*(G(q)*H(q))^6).

%K nonn

%O 0,7

%A _Joerg Arndt_, Oct 07 2012