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A294467
Binomial transform of A088311.
8
1, 2, 5, 22, 113, 746, 6037, 55070, 548417, 6281938, 79935941, 1087584422, 16109401585, 255667890362, 4358283982613, 79893373511086, 1542859916102657, 31322024816838050, 676027617881188357, 15287136167625123638, 362322855217463741681
OFFSET
0,2
LINKS
FORMULA
a(n) ~ exp(1) * n! * A000009(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(n/3) - n + 1) * n^(n - 1/4) / (4*3^(1/4)).
E.g.f.: exp(x) * Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Oct 15 2018
MATHEMATICA
Table[Sum[Binomial[n, k]*k!*PartitionsQ[k], {k, 0, n}], {n, 0, 20}]
PROG
(PARI) x='x+O('x^30); Vec(serlaplace(exp(x)*eta(x^2)/eta(x))) \\ G. C. Greubel, Oct 15 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)*(&*[1 + x^k: k in [1..50]]))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 15 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 31 2017
STATUS
approved