A299108 - OEIS

%I #16 Mar 31 2018 17:59:37

%S 1,1,3,9,27,79,231,675,1971,5755,16805,49071,143289,418411,1221781,

%T 3567663,10417761,30420401,88829145,259385701,757419669,2211704625,

%U 6458291945,18858546645,55067931981,160801210705,469547855419,1371104033121,4003694720243

%N Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).

%H Vaclav Kotesovec, <a href="/A299108/b299108.txt">Table of n, a(n) for n = 0..2000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).

%F G.f.: 1/(1 - x/theta_4(x)), where theta_4() is the Jacobi theta function.

%F a(0) = 1; a(n) = Sum_{k=1..n} A015128(k-1)*a(n-k).

%F a(n) ~ c * d^n, where d = 2.9200517419026569743994130834319365190407162724411912701937027582419975778... is the root of the equation EllipticTheta(4, 0, 1/d) * d = 1 and c = 0.372842695601022868809531452599286285949969156503576039087883242107... = 2*Log[r]*QPochhammer[r] / (2*QPochhammer[r] * (Log[1 - r] + Log[r] + QPolyGamma[1, r]) + r*Log[r] * (r * Derivative[0, 1][QPochhammer][-1, r] - 2*Derivative[0, 1][QPochhammer][r, r])), where r = 1/d. Equivalently, c = EllipticTheta[4, 0, r]^2 / (r *(EllipticTheta[4, 0, r] - r * Derivative[0, 0, 1][EllipticTheta][4, 0, r])). - _Vaclav Kotesovec_, Feb 03 2018, updated Mar 31 2018

%p S:= series(1/(1-x/JacobiTheta4(0,x)),x,51):

%p seq(coeff(S,x,n),n=0..50); # _Robert Israel_, Feb 02 2018

%t nmax = 28; CoefficientList[Series[1/(1 - x Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%t nmax = 28; CoefficientList[Series[1/(1 - x/EllipticTheta[4, 0, x]), {x, 0, nmax}], x]

%t nmax = 28; CoefficientList[Series[1/(1 - x QPochhammer[-x, x]/QPochhammer[x, x]), {x, 0, nmax}], x]

%Y Antidiagonal sums of A288515.

%Y Cf. A015128, A032803, A067687, A299105, A299106.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Feb 02 2018