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A217569 Expansion of H(q)*G(q^11) where H and G are respectively the g.f. of A003106 and A003114 (Rogers-Ramanujan functions). 0
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 (list; graph; refs; listen; history; text; internal format)
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 * A026823 A025148 A096749

Adjacent sequences:  A217566 A217567 A217568 * A217570 A217571 A217572

KEYWORD

nonn

AUTHOR

Joerg Arndt, Oct 07 2012

STATUS

approved

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Last modified May 27 05:48 EDT 2020. Contains 334649 sequences. (Running on oeis4.)