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A036018 Number of partitions of n into parts not of form 4k+2, 12k, 12k+3 or 12k-3.

%I A036018 #31 Feb 16 2025 08:32:37
%S A036018 1,1,1,1,2,3,3,4,6,7,8,10,13,16,18,22,28,33,38,45,55,65,74,87,104,121,
%T A036018 138,160,188,217,247,284,330,378,428,489,562,640,722,820,936,1059,
%U A036018 1191,1345,1524,1717,1924,2163,2438,2734,3054,3419,3834,4284,4770,5321,5943
%N A036018 Number of partitions of n into parts not of form 4k+2, 12k, 12k+3 or 12k-3.
%C A036018 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A036018 Case k=3,i=2 of Gordon/Goellnitz/Andrews Theorem.
%C A036018 Also number of partitions in which no odd part is repeated, with at most 1 part of size less than or equal to 2 and where differences between parts at distance 2 are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.
%D A036018 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
%H A036018 Vaclav Kotesovec, <a href="/A036018/b036018.txt">Table of n, a(n) for n = 0..10000</a>
%H A036018 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects of partitions</a>, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
%H A036018 Andrew Sills, <a href="http://home.dimacs.rutgers.edu/~asills/EMDC/SillsEMDC-Rev.pdf">Towards an Automation of the Circle Method</a>, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S27.
%H A036018 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A036018 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A036018 Expansion of q^(-1/4) * (eta(q^2) * eta(q^3) * eta(q^12)) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q. - _Michael Somos_, Jun 28 2004
%F A036018 Euler transform of period 12 sequence [1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, ...]. - _Michael Somos_, Jun 28 2004
%F A036018 Expansion of psi(-x^3) / psi(-x) in powers of x where psi() is a Ramanujan theta function. - _Michael Somos_, Nov 21 2007
%F A036018 Given g.f. A(x), then B(x) = x * A(x^4) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = u^3 * (1 + v^4) - v * (1 + u*v)^3. - _Michael Somos_, Nov 21 2007
%F A036018 G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = sqrt(1 / 3) / f(t) where q = exp(2 Pi i t). - _Michael Somos_, Nov 21 2007
%F A036018 A101195(n) = (-1)^n * a(n).
%F A036018 From _Vaclav Kotesovec_, Jan 12 2017: (Start)
%F A036018 a(n) ~ Pi * BesselI(1, sqrt(4*n+1)*Pi/(2*sqrt(3))) / (3*sqrt(4*n+1)).
%F A036018 a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)) * (1 + (Pi/24 - 3/(8*Pi))*sqrt(3/n) + (Pi^2/384 - 45/(128*Pi^2) - 15/64)/n).
%F A036018 (End)
%e A036018 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
%e A036018 q + q^5 + q^9 + q^13 + 2*q^17 + 3*q^21 + 3*q^25 + 4*q^29 + 6*q^33 + ...
%t A036018 nmax = 50; CoefficientList[Series[Product[(1 + x^(6*k - 5))*(1 + x^(6*k - 1))/((1 - x^(6*k - 4))*(1 - x^(6*k - 2))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 11 2017 *)
%o A036018 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ([1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0][(k-1)%12 + 1]) * x^k, 1 + x * O(x^n)), n))} /* _Michael Somos_, Jun 28 2004 */
%o A036018 (PARI) {a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))} /* _Michael Somos_, Jun 28 2004 */
%Y A036018 Cf. A101195.
%K A036018 nonn,easy
%O A036018 0,5
%A A036018 _Olivier Gérard_