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A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.

%I A101277 #46 Feb 16 2025 08:32:55
%S A101277 1,2,3,6,10,16,25,38,57,84,121,172,243,338,465,636,862,1158,1546,2050,
%T A101277 2702,3542,4616,5986,7729,9932,12707,16196,20563,26010,32788,41194,
%U A101277 51591,64418,80195,99558,123269,152226,187514,230434,282519,345596,421844,513834
%N A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.
%C A101277 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A101277 This is also A080054 times 1/Product_{k>=1} (1 - x^(2k)).
%C A101277 There are no partitions of 2n+1 in which all odd parts occur with multiplicity 2. - _Michael Somos_, Oct 27 2008
%H A101277 G. C. Greubel, <a href="/A101277/b101277.txt">Table of n, a(n) for n = 0..1000</a>
%H A101277 Cristina Ballantine, Mircea Merca, <a href="https://doi.org/10.1007/s00009-019-1301-6">Jacobi's Four and Eight Squares Theorems and Partitions into Distinct Parts</a>, Mediterranean Journal of Mathematics (2019) Vol. 16, No. 2, 26.
%H A101277 Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.
%H A101277 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 A101277 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A101277 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A101277 Euler transform of period 4 sequence [2, 0, 2, 1, ...]. - _Michael Somos_, Feb 10 2005
%F A101277 G.f.: (1/theta_4(0, x))*Product_{k>0}(1+x^(2k)) = theta_4(0, x^2)/theta_4(0, x)*Product_{k>0}(1-x^(2k)) = 1/Product_{k>0} ((1-x^(2k-1))^2 * (1-x^(4k))).
%F A101277 Expansion of 1 / (psi(-x) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - _Michael Somos_, Oct 27 2008
%F A101277 Expansion of q^(1/12) * eta(q^2)^2 / (eta(q)^2 * eta(q^4)) in powers of q. - _Michael Somos_, Oct 27 2008
%F A101277 a(n) ~ sqrt(5) * exp(Pi*sqrt(5*n/6)) / (8*sqrt(3)*n). - _Vaclav Kotesovec_, Aug 30 2015
%F A101277 G.f.: 2/((x; x)_inf * (-1; -x)_inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Nov 22 2016
%F A101277 Expansion of phi(-x^2) / f(-x)^2 = chi(x) / f(-x) = 1 / (chi(-x)^2 * f(-x^4)) = f(-x^4) / psi(-x)^2 = psi(-x) / chi(-x) = chi(x)^2 / psi(-x^2) in powers of x. - _Michael Somos_, Nov 22 2016
%e A101277 G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 57*x^8 + ...
%e A101277 G.f. = 1/q + 2*q^11 + 3*q^23 + 6*q^35 + 10*q^47 + 16*q^59 + 25*q^71 + ...
%e A101277 E.g. 12 = 10 + 2 = 10 + 1 + 1 = 8 + 4 = 8 + 2 + 2 = 8 + 2 + 1 + 1 = 6 + 6 = 6 + 4 + 2 = 6 + 4 + 1 + 1 = 6 + 3 + 3 = 6 + 2 + 2 + 2 = 6 + 2 + 2 + 1 + 1 = 5 + 5 + 2 = 5 + 5 + 1 + 1 = 4 + 4 + 4 = 4 + 4 + 2 + 2 = 4 + 4 + 2 + 1 + 1 = 4 + 3 + 3 + 2 = 4 + 3 + 3 + 1 + 1 = 4 + 2 + 2 + 2 + 2 = 4 + 2 + 2 + 2 + 1 + 1 = 3 + 3 + 2 + 2 + 2 = 3 + 3 + 2 + 2 + 1 + 1 = 2 + 2 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 + 1 + 1.
%p A101277 series(product(1/((1-x^(2*k-1))^2*(1-x^(4*k))),k=1..100),x=0,100);
%t A101277 nmax=50; CoefficientList[Series[Product[1/((1-x^(2*k-1))^2 * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 30 2015 *)
%t A101277 (2/(QPochhammer[x] QPochhammer[-1, -x]) + O[x]^45)[[3]] (* _Vladimir Reshetnikov_, Nov 22 2016 *)
%t A101277 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] / QPochhammer[ x]^2, {x, 0, n}]; (* _Michael Somos_, Nov 22 2016 *)
%t A101277 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Nov 22 2016 *)
%t A101277 a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, -x] QPochhammer[ x]), {x, 0, n}]; (* _Michael Somos_, Nov 22 2016 *)
%o A101277 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* _Michael Somos_, Feb 10 2005 */
%Y A101277 Cf. A015128, A098151, A080054.
%K A101277 nonn
%O A101277 0,2
%A A101277 _Noureddine Chair_, Dec 20 2004; revised Jan 05 2005