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 A299108 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)). 16
 1, 1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16805, 49071, 143289, 418411, 1221781, 3567663, 10417761, 30420401, 88829145, 259385701, 757419669, 2211704625, 6458291945, 18858546645, 55067931981, 160801210705, 469547855419, 1371104033121, 4003694720243 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 N. J. A. Sloane, Transforms FORMULA G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)). G.f.: 1/(1 - x/theta_4(x)), where theta_4() is the Jacobi theta function. a(0) = 1; a(n) = Sum_{k=1..n} A015128(k-1)*a(n-k). 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 MAPLE S:= series(1/(1-x/JacobiTheta4(0, x)), x, 51): seq(coeff(S, x, n), n=0..50); # Robert Israel, Feb 02 2018 MATHEMATICA nmax = 28; CoefficientList[Series[1/(1 - x Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] nmax = 28; CoefficientList[Series[1/(1 - x/EllipticTheta[4, 0, x]), {x, 0, nmax}], x] nmax = 28; CoefficientList[Series[1/(1 - x QPochhammer[-x, x]/QPochhammer[x, x]), {x, 0, nmax}], x] CROSSREFS Antidiagonal sums of A288515. Cf. A015128, A032803, A067687, A299105, A299106. Sequence in context: A027129 A077844 A077827 * A304067 A287898 A129770 Adjacent sequences: A299105 A299106 A299107 * A299109 A299110 A299111 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Feb 02 2018 STATUS approved

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Last modified March 29 01:21 EDT 2023. Contains 361596 sequences. (Running on oeis4.)