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%I #27 Apr 18 2021 02:36:48
%S 1,1,2,3,5,7,9,13,18,24,32,42,55,71,91,116,147,185,231,288,357,440,
%T 540,661,807,980,1186,1432,1724,2069,2476,2956,3521,4183,4958,5865,
%U 6923,8155,9587,11251,13180,15411,17990,20967,24399,28348,32886,38098,44075
%N Expansion of phi(-x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Partitions which are "4-close" to being self-conjugate; see A108961 for the definition. - _Arvind Ayyer_, Apr 13 2021
%D D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988. See page 6 equation 2.
%H Vaclav Kotesovec, <a href="/A271661/b271661.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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey_pair">Bailey pair</a>
%F Expansion of f(x^2, x^4) / psi(-x) in powers of x where psi(), f(, ) are Ramanujan theta functions.
%F Expansion of q^(1/24) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.
%F Euler transform of period 12 sequence [ 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 0, ...].
%F G.f.: Sum_{k>=0} x^(k^2) (-x, x^2)_k / (x)_{2*k}.
%F a(n) ~ Pi * BesselI(1, Pi*sqrt(24*n-1)/(4*sqrt(3))) / sqrt(24*n-1) ~ exp(sqrt(n/2)*Pi) / (2^(7/4)*sqrt(3)*n^(3/4)) * (1 - (3/(4*Pi) + Pi/48)/sqrt(2*n) + (5/128 - 15/(64*Pi^2) + Pi^2/9216)/n). - _Vaclav Kotesovec_, Apr 18 2016, extended Jan 10 2017
%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 13*x^7 + 18*x^8 + ...
%e G.f. = q^-1 + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 9*q^143 + 13*q^167 + ..
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^6] / QPochhammer[ x], {x, 0, n}];
%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x, x, 2 k] // FunctionExpand, {k, 0, Sqrt@n}], {x, 0, n}]];
%t nmax = 50; CoefficientList[Series[Product[(1-x^(6*k)) / ((1-x^k) * (1+x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 18 2016 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x + A) * eta(x^12 + A)), n))};
%Y Cf. A000700, A070047, A108961, A108962, A280937, A280938.
%K nonn
%O 0,3
%A _Michael Somos_, Apr 11 2016
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