OFFSET
0,3
COMMENTS
Let (a1,a2,a3,...ad:b1,b2,b3,...bd) be the Frobenius symbol for a partition pi of n. Then pi is m-close to being self-conjugate if for all k, |ak-bk| <= m.
REFERENCES
D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
D. M. Bressoud, Extension of the partition sieve, J. Number Theory 12 no. 1 (1980), 87-100.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Wikipedia, Bailey pair
FORMULA
Define the Dedekind eta function = q^1/24. Product(1-q^k), k >=1. Then the number of m-close partitions is q^(1/24).(m+2)^2/(1.(2m+4)) (where m denotes eta(q^m)).
From Vaclav Kotesovec, Nov 13 2016, extended Jan 11 2017: (Start)
a(n, m) ~ exp(Pi*sqrt((2*m+1)*n/(3*(m+2)))) * (2*m+1)^(1/4) / (2*3^(1/4)*(m+2)^(3/4)*n^(3/4)).
For m=3, a(n) ~ 7^(1/4) * exp(sqrt(7*n/15)*Pi) / (2*3^(1/4)*5^(3/4)*n^(3/4)) * (1 -(3*sqrt(15)/(8*Pi*sqrt(7)) + Pi*sqrt(7)/(48*sqrt(15)))/sqrt(n) + (7*Pi^2/69120 - 225/(896*Pi^2) + 5/128)/n).
(End)
a(n) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)*7/10)) / (5*sqrt((24*n-1)/7)). - Vaclav Kotesovec, Jan 11 2017
EXAMPLE
1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 11*x^7 + 16*x^8 + 20*x^9 + 28*x^10 + ...
1/q + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 5*q^119 + 9*q^143 + 11*q^167 + 16*q^191 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^2 / ((1 - x^k) * (1 - x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A)^2 / (eta(x + A) * eta(x^10 + A)), n))} /* Michael Somos, Jun 08 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005
STATUS
approved