A108962 Number of partitions that are "3-close" to being self-conjugate.

1, 1, 2, 3, 5, 5, 9, 11, 16, 20, 28, 34, 47, 57, 75, 92, 119, 143, 183, 220, 277, 332, 412, 491, 605, 718, 874, 1036, 1252, 1475, 1772, 2082, 2483, 2909, 3450, 4027, 4755, 5533, 6499, 7545, 8826, 10213, 11904, 13741, 15955, 18372, 21262, 24422

Offset: 0

Author

John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005

Keywords

• nonn

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.

Examples

			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 + ...
		

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

Crossrefs

Cf. A000700 for m=0 (self-conjugate), A070047 for m=1, A108961 for m=2, A271661 for m=4, A280937 for m=5, A280938 for m=6.

Programs

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 *)

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 */

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