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%I #23 Apr 18 2021 02:36:18
%S 1,1,2,3,5,5,9,11,16,20,28,34,47,57,75,92,119,143,183,220,277,332,412,
%T 491,605,718,874,1036,1252,1475,1772,2082,2483,2909,3450,4027,4755,
%U 5533,6499,7545,8826,10213,11904,13741,15955,18372,21262,24422
%N Number of partitions that are "3-close" to being self-conjugate.
%C 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.
%D D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
%H Vaclav Kotesovec, <a href="/A108962/b108962.txt">Table of n, a(n) for n = 0..1000</a>
%H D. M. Bressoud, <a href="https://doi.org/10.1016/0022-314X(80)90077-3">Extension of the partition sieve</a>, J. Number Theory 12 no. 1 (1980), 87-100.
%H Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey_pair">Bailey pair</a>
%F 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)).
%F From _Vaclav Kotesovec_, Nov 13 2016, extended Jan 11 2017: (Start)
%F 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)).
%F 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).
%F (End)
%F 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
%e 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 + ...
%e 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 + ...
%t 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 *)
%o (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 */
%Y 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.
%K nonn
%O 0,3
%A John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005
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