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A131945 Number of partitions of n where odd parts are distinct or repeated once.

%I A131945 #44 Feb 16 2025 08:33:06
%S A131945 1,1,2,2,4,5,8,10,15,18,26,32,45,55,74,90,119,145,188,228,291,351,442,
%T A131945 532,664,796,982,1172,1435,1708,2076,2462,2972,3512,4214,4966,5929,
%U A131945 6965,8272,9688,11457,13383,15762,18362,21543,25031,29264,33922,39533,45717
%N A131945 Number of partitions of n where odd parts are distinct or repeated once.
%C A131945 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A131945 Also number of partitions of n such that every part is not congruent to 3 mod 6. More generally, g.f. for number of partitions of n such that every odd part occurs at most m times is product_{n=1..oo} (1-q^((m+1)*(2*n-1)))/(1-q^n). Similarly, g.f. for number of partitions of n such that every even part occurs at most m times is product_{n=1..oo} (1-q^((2*m+2)*n))/(1-q^n). - _Vladeta Jovovic_, Aug 01 2007
%H A131945 Brian Drake and Seiichi Manyama, <a href="/A131945/b131945.txt">Table of n, a(n) for n = 0..1000</a> (first 101 terms from Brian Drake)
%H A131945 Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953.
%H A131945 Mircea Merca, <a href="https://doi.org/10.1007/s00010-024-01117-6">Overpartitions in terms of 2-adic valuation</a>, Aequat. Math. (2024). See p. 11.
%H A131945 James A. Sellers, <a href="https://arxiv.org/abs/2409.12321">Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice</a>, arXiv:2409.12321 [math.NT], 2024. See p. 2.
%H A131945 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A131945 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A131945 G.f.: product_{n=1..oo} (1-q^(6n-3))/(1-q^n).
%F A131945 Expansion of chi(-x^3) / f(-x) in powers of x where chi(), f() are Ramanujan theta functions. - _Michael Somos_, Aug 05 2007
%F A131945 Expansion of q^(1/6) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - _Michael Somos_, Aug 05 2007
%F A131945 Euler transform of period 6 sequence [ 1, 1, 0, 1, 1, 1, ...]. - _Michael Somos_, Aug 05 2007
%F A131945 a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (12*n). - _Vaclav Kotesovec_, Dec 11 2016
%e A131945 a(6) = 8 because we have 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 2+2+2 and 2+2+1+1. The three excluded partitions of 6 are 3+1+1+1, 2+1+1+1+1 and 1+1+1+1+1+1.
%e A131945 G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ...
%e A131945 G.f. = 1/q + q^5 + 2*q^11 + 2*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 10*q^41 + ...
%p A131945 A:= series(product( (1-q^(6*n-3))/(1-q^n), n=1..20),q,21): seq(coeff(A,q,i), i=0..20);
%t A131945 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}]; (* _Michael Somos_, Jan 29 2015 *)
%t A131945 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)
%t A131945 nmax = 50; CoefficientList[Series[Product[1 / ((1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 11 2016 *)
%o A131945 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* _Michael Somos_, Aug 05 2007 */
%Y A131945 Cf. A000122, A000700, A006950, A010054, A121373, A131942, A279320.
%K A131945 easy,nonn
%O A131945 0,3
%A A131945 _Brian Drake_, Jul 30 2007