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 A333152 G.f.: Sum_{k>=1} (k^4 * x^(k^2) / Product_{j=1..k} (1 - x^j)). 3
 0, 1, 1, 1, 17, 17, 33, 33, 49, 130, 146, 227, 324, 405, 502, 664, 1017, 1179, 1613, 2031, 2721, 3220, 4166, 4921, 6204, 7840, 9379, 11352, 14028, 16882, 20520, 24511, 29286, 34864, 41401, 48741, 58417, 68144, 80207, 93698, 110325, 128124, 150436, 173424 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS In general, if m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k^2) / Product_{j=1..k} (1 - x^j)), then a(n) ~ r^m * phi^(1/2) * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2)), where r = A333155 = sqrt(15) * log(phi) / Pi = 0.59324221500336912718413761733... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 FORMULA a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(5/4), where c = A333155^4 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.026768664197762321048783840410317... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))) end: a:= n-> add(k^4 * b(n-k^2, k), k=1..floor(sqrt(n))): seq(a(n), n=0..50); # after Alois P. Heinz MATHEMATICA nmax = 50; CoefficientList[Series[Sum[n^4*x^(n^2)/Product[1-x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x] CROSSREFS Cf. A003114, A268188, A333141, A333151. Sequence in context: A081702 A345735 A238235 * A040273 A022351 A146873 Adjacent sequences: A333149 A333150 A333151 * A333153 A333154 A333155 KEYWORD nonn AUTHOR Vaclav Kotesovec, Mar 09 2020 STATUS approved

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Last modified June 1 02:02 EDT 2023. Contains 363068 sequences. (Running on oeis4.)