A217569 Expansion of H(q)*G(q^11) where H and G are respectively the g.f. of A003106 and A003114 (Rogers-Ramanujan functions).

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, 72, 81, 94, 105, 121, 136, 155, 175, 199, 222, 252, 282, 318, 355, 400, 445, 501, 556, 624, 693, 774, 857, 957, 1059, 1178, 1302, 1446, 1596, 1769, 1951, 2158, 2376, 2624, 2885, 3182, 3495, 3847, 4221, 4642

Offset: 0

Author

Joerg Arndt, Oct 07 2012

Keywords

• nonn

Comments

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.

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}.

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.

Links

• Alexander Berkovich, Hamza Yesilyurt, On Rogers-Ramanujan functions, binary quadratic forms and eta-quotients, arXiv:1204.1092v2 [math.NT], 2012.
• Srinivasa Ramanujan, Algebraic relations between certain infinite products, Proceedings of the London Mathematical Society, vol.2, no.18, 1920.

Crossrefs

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

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

Programs

PARI

N=66; q='q+O('q^N );
S=2+2*ceil(sqrt(N));
G(q)=sum(n=0,S,q^(n^2)/prod(k=1,n,1-q^k)); /* g.f. of  A003114 */
H(q)=sum(n=0,S,q^(n^2+n)/prod(k=1,n,1-q^k)); /* g.f. of A003106 */
Vec(H(q)*G(q^11)) /* show terms */
/* checking the modular equations, all expressions are zero:
( H(q)*G(q)^11 - q^2*G(q)*H(q)^11 ) - ( 1 + 11*q*(G(q)*H(q))^6 )
( H(q)*G(q^11) - q^2*G(q)*H(q^11) ) - ( 1 )
E(q)=prod(n=1,N, 1-q^n);
G(q)*H(q) - E(q^5)/E(q)
G(q) - ( E(q^8)/E(q^2) * (G(q^16) + q*H(-q^4)) )
H(q) - ( E(q^8)/E(q^2) * (q^3*H(q^16) + G(-q^4)) )
*/

PARI

N=66; q='q+O('q^N );
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];
Vec( 1 / prod(K=0, N\55+1,  prod(k=1,24, 1 - q^(K*55+E[k]) ) ) )

Formula

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).

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).