%I #32 Mar 12 2021 22:24:43

%S 1,1,3,4,7,10,17,23,35,48,69,93,131,173,236,310,413,536,704,903,1170,

%T 1489,1904,2403,3044,3811,4784,5951,7409,9157,11325,13912,17095,20891,

%U 25519,31029,37708,45632,55184,66495,80050,96064,115173,137680,164425,195860

%N Expansion of psi(x^2) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C This sequence convolved with A000009 gives A001936. - _Gary W. Adamson_, Mar 24 2011

%C a(n) is the number of partitions of n in which each odd part can occur any number of times but each even part is of two kinds and each kind can occur at most once. - _Michael Somos_, Dec 01 2019

%H G. C. Greubel, <a href="/A098613/b098613.txt">Table of n, a(n) for n = 0..1000</a>

%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>

%F Expansion of chi(x) / chi(-x^2)^3 = 1 / (chi(-x)* chi(-x^2)^2) = 1 / (chi(x)^2 * chi(-x)^3) in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Sep 07 2015

%F Expansion of q^(-5/24) * eta(q^4)^2 / (eta(q) * eta(q^2)) in powers of q.

%F Euler transform of period 4 sequence [1, 2, 1, 0, ...].

%F G.f. A(x) is the limit of x^(n^2+n) * P_{2*n+1}(1/x)/2 where P_n(q) = Sum_{k=0..n} C(n, k; q) and C(n, k; q) is the q-binomial coefficients. See A083906 for P_n(q).

%F G.f.: (Sum_{k>0} x^(k^2-k)) / (Product_{k>0} (1 - x^k)).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 8^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143161. - _Michael Somos_, Sep 07 2015

%F G.f.: Product_{k>=1} (1 + x^(2*k))^2 / (1 - x^(2*k-1)). - _Michael Somos_, Dec 01 2019

%e G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 23*x^7 + ...

%e G.f. = q^5 + q^29 + 3*q^53 + 4*q^77 + 7*q^101 + 10*q^125 + 17*q^149 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]), {x, 0, n}]; (* _Michael Somos_, Oct 29 2013 *)

%t a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[ x^2, x^4]^2), {x, 0, n}]; (* _Michael Somos_, Oct 29 2013 *)

%t nmax = 40; CoefficientList[ Series[Product[(1 + x^k) * (1 + x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^2]^3, {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x]^2 QPochhammer[ -x, x]^3, {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^2, x^2]^2, {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1)-1)\2, x^(k^2+k)) / eta(x + x * O(x^n)), n))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)), n))};

%o (PARI) {a(n) = if( n<0,0, polcoeff( sum(k=0, 2*n+1, prod(i=1, k, (1 - x^(2*n+2 - i)) / (1 - x^i))) / 2, n^2))};

%Y Cf. A029552, A083906, A143161.

%K nonn

%O 0,3

%A _Michael Somos_, Sep 17 2004