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

%I A108961 #36 Feb 16 2025 08:32:58
%S A108961 1,1,2,3,3,5,7,9,12,16,20,26,33,41,51,64,79,97,119,144,175,212,254,
%T A108961 305,365,434,516,612,722,851,1002,1174,1375,1607,1872,2179,2531,2933,
%U A108961 3395,3923,4524,5211,5994,6881,7891,9038,10334,11804,13467,15341,17460,19849
%N A108961 Number of partitions that are "2-close" to being self-conjugate.
%C A108961 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.
%C A108961 Convolution of A070048 and A035457. - _Vaclav Kotesovec_, Nov 13 2016
%D A108961 D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
%H A108961 Vaclav Kotesovec, <a href="/A108961/b108961.txt">Table of n, a(n) for n = 0..1000</a>
%H A108961 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 A108961 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 A108961 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A108961 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A108961 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey_pair">Bailey pair</a>
%F A108961 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 A108961 Expansion of q^(1/24) * eta(q^4)^2 / (eta(q) * eta(q^8)) in powers of q. - _Michael Somos_, Oct 17 2006
%F A108961 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]
%F A108961 Euler transform of period 8 sequence [ 1, 1, 1, -1, 1, 1, 1, 0, ...]. - _Michael Somos_, Oct 17 2006
%F A108961 G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k)) / (1 + x^(4*k)). - _Michael Somos_, Oct 17 2006
%F A108961 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
%e A108961 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 + ...
%e A108961 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 + ...
%t A108961 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 *)
%o A108961 (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 */
%Y A108961 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.
%K A108961 nonn
%O A108961 0,3
%A A108961 John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005