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

0,7

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

Table of n, a(n) for n=0..66.

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.

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).

Prog

(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]) ) ) )

Crossrefs

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

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

Sequence in context: A046676 A003114 A185227 * A335766 A026823 A025148

Adjacent sequences: A217566 A217567 A217568 * A217570 A217571 A217572

Keyword

nonn

Author

Joerg Arndt, Oct 07 2012

Status

approved