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 A261569 Expansion of Product_{k>=1} (1 + 5*x^k). 5
 1, 5, 5, 30, 30, 55, 180, 205, 330, 480, 1230, 1380, 2255, 3030, 4530, 8555, 10680, 15330, 21330, 29730, 39480, 67380, 81505, 116280, 153030, 210930, 270805, 370080, 534330, 675480, 900480, 1180380, 1544130, 1997280, 2597280, 3304805, 4581180, 5653080 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 FORMULA a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(6*Pi)*n^(3/4)), where c = Pi^2/6 + log(5)^2/2 + polylog(2, -1/5) = 2.74927912606080829002558751537626864449... . - Vaclav Kotesovec, Jan 04 2016 G.f.: Sum_{i>=0} 5^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018 MAPLE b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 5*b(n-i, i-1))))     end: a:= n-> b(n\$2): seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015 MATHEMATICA nmax = 40; CoefficientList[Series[Product[1 + 5*x^k, {k, 1, nmax}], {x, 0, nmax}], x] nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*5^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *) (QPochhammer[-5, x]/6 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *) CROSSREFS Cf. A000009, A032302, A032308, A261568, A291698. Sequence in context: A284182 A020551 A153271 * A117858 A014434 A106830 Adjacent sequences:  A261566 A261567 A261568 * A261570 A261571 A261572 KEYWORD nonn AUTHOR Vaclav Kotesovec, Aug 24 2015 STATUS approved

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Last modified December 14 05:36 EST 2019. Contains 329978 sequences. (Running on oeis4.)