A108961 Number of partitions that are "2-close" to being self-conjugate.

1, 1, 2, 3, 3, 5, 7, 9, 12, 16, 20, 26, 33, 41, 51, 64, 79, 97, 119, 144, 175, 212, 254, 305, 365, 434, 516, 612, 722, 851, 1002, 1174, 1375, 1607, 1872, 2179, 2531, 2933, 3395, 3923, 4524, 5211, 5994, 6881, 7891, 9038, 10334, 11804, 13467, 15341, 17460, 19849

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.

Convolution of A070048 and A035457. - Vaclav Kotesovec, Nov 13 2016

Examples

			1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 12*x^8 + 16*x^9 + 20*x^10 + ...
1/q + q^23 + 2*q^47 + 3*q^71 + 3*q^95 + 5*q^119 + 7*q^143 + 9*q^167 + 12*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.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Wikipedia, Bailey pair

Crossrefs

Cf. A000700 for m=0 (self-conjugate), A070047 for m=1, A108962 for m=3, A271661 for m=4, A280937 for m=5, A280938 for m=6.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(2*k)) / (1 + x^(4*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^4 + A)^2 / (eta(x + A) * eta(x^8 + A)), n))} /* Michael Somos, Oct 17 2006 */

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)).

Expansion of q^(1/24) * eta(q^4)^2 / (eta(q) * eta(q^8)) in powers of q. - Michael Somos, Oct 17 2006

Expansion of chi(x^2) * chi(x) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function (see A000700). - Michael Somos, Oct 17 2006 [corrected by Peter Bala, Oct 09 2023]

Euler transform of period 8 sequence [ 1, 1, 1, -1, 1, 1, 1, 0, ...]. - Michael Somos, Oct 17 2006

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k)) / (1 + x^(4*k)). - Michael Somos, Oct 17 2006

a(n) ~ Pi * BesselI(1, Pi * sqrt(5*(24*n-1)/2)/12) / (2*sqrt((24*n-1)/5)) ~ 5^(1/4) * exp(sqrt(5*n/3)*Pi/2) / (2^(5/2) * 3^(1/4) * n^(3/4)) * (1 -(3*sqrt(3) / (4*Pi*sqrt(5)) + Pi*sqrt(5)/(96*sqrt(3)))/sqrt(n) + (5*Pi^2/55296 - 9/(32*Pi^2) + 5/128)/n). - Vaclav Kotesovec, Nov 13 2016, extended Jan 11 2017