# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a330505 Showing 1-1 of 1 %I A330505 #6 Dec 16 2019 14:25:37 %S A330505 1,2,8,24,144,960,5760,40320,524160,4354560,43545600,638668800, %T A330505 6706022400,99632332800,2092278988800,20922789888000,376610217984000, %U A330505 9247873130496000,128047474114560000,2919482409811968000,77852864261652480000 %N A330505 Expansion of e.g.f. Sum_{k>=1} arctanh(x^k). %F A330505 E.g.f.: -log(theta_4(x)) / 2. %F A330505 E.g.f.: (1/2) * Sum_{k>=1} log((1 + x^k) / (1 - x^k)). %F A330505 E.g.f.: log(Product_{k>=1} ((1 + x^k) / (1 - x^k))^(1/2)). %F A330505 E.g.f.: Sum_{k>=1} x^(2*k - 1) / ((2*k - 1) * (1 - x^(2*k - 1))). %F A330505 exp(2 * Sum_{n>=1} a(n) * x^n / n!) = g.f. of A015128. %F A330505 a(n) = (n - 1)! * Sum_{d|n, n/d odd} d. %t A330505 nmax = 21; CoefficientList[Series[Sum[ArcTanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest %t A330505 nmax = 21; CoefficientList[Series[-Log[EllipticTheta[4, 0, x]]/2, {x, 0, nmax}], x] Range[0, nmax]! // Rest %t A330505 Table[(n - 1)! DivisorSum[n, # &, OddQ[n/#] &], {n, 1, 21}] %Y A330505 Cf. A002131, A002448, A010050, A015128, A015680, A038048, A054785, A228274, A265024, A330504. %K A330505 nonn %O A330505 1,2 %A A330505 _Ilya Gutkovskiy_, Dec 16 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE