A332258 - OEIS

%I #8 Feb 08 2020 20:41:17

%S 1,0,0,1,0,1,20,1,112,1681,492,27721,371624,319177,13461604,171387217,

%T 319071456,11466038689,143550642140,484491620089,15758152572952,

%U 199089883272217,1077471975974484,32827750137627457,427744154995090256,3385134777669637681

%N E.g.f.: 1 / (1 + x - sinh(x)).

%C Number of labeled ordered partitions of an n-set into odd parts > 1.

%F a(0) = 1; a(n) = Sum_{k=2..ceiling(n/2)} binomial(n,2*k-1) * a(n-2*k+1).

%F a(n) ~ n! / ((cosh(r) - 1) * r^(n+1)), where r = 1.72911689821437486498840709347... is the root of the equation 1 + r - sinh(r) = 0. - _Vaclav Kotesovec_, Feb 08 2020

%t nmax = 25; CoefficientList[Series[1/(1 + x - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k - 1] a[n - 2 k + 1], {k, 2, Ceiling[n/2]}]; Table[a[n], {n, 0, 25}]

%o (PARI) seq(n)={Vec(serlaplace(1 / (1 + x - sinh(x + O(x*x^n)))))} \\ _Andrew Howroyd_, Feb 08 2020

%Y Cf. A006154, A009227, A032032.

%K nonn

%O 0,7

%A _Ilya Gutkovskiy_, Feb 08 2020