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a(n) ~ c * d^n / sqrt(n), where d = 4.3505530790182509701639869563721679988879373943131559534408716195123... and c = 0.2276354216252041005336767937139336687746108521151301186102034... - Vaclav Kotesovec, Aug 18 2018

Vaclav Kotesovec, <a href="/A318118/b318118.txt">Table of n, a(n) for n = 0..1000</a>

proposed

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allocated for Ilya Gutkovskiy

a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

1, 1, 3, 10, 40, 150, 616, 2456, 10102, 41400, 171526, 712111, 2972115, 12434993, 52195414, 219567909, 925704792, 3909841659, 16541598215, 70085877919, 297347922785, 1263046810334, 5370930049915, 22861883482838, 97402827429118, 415332438952517, 1772380322197432

0,3

For n > 2, a(n) is the n-th term of the Euler transform of n-gonal numbers.

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

<a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

Cf. A000294, A023871, A274998, A278767, A278768, A278769.

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nonn

Ilya Gutkovskiy, Aug 18 2018

approved

editing

allocated for Ilya Gutkovskiy

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approved