A111752 Number of partitions of {1,..,n} into lists with an even number of lists of size 1, where a list means an ordered subset (cf. A000262).
1, 0, 3, 6, 49, 300, 2491, 22890, 239457, 2782584, 35595091, 496577070, 7499663953, 121855323876, 2118793593099, 39245026343250, 771255810671041, 16025261292247920, 350956070419872547, 8078570913162379734, 194969375055353840241, 4922311437793379501340
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..444
Programs
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Maple
b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j, `if`(j=1, 1-t, t))*binomial(n-1, j-1)*j!, j=1..n)) end: a:= n-> b(n, 1): seq(a(n), n=0..30); # Alois P. Heinz, May 10 2016 -
Mathematica
b[n_, t_] := b[n, t] = If[n == 0, t, Sum[b[n-j, If[j == 1, 1-t, t]] * Binomial[n-1, j-1]*j!, {j, 1, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *) -
Python
from sympy.core.cache import cacheit from sympy import binomial, factorial as f @cacheit def b(n, t): return t if n==0 else sum(b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1)*f(j) for j in range(1, n + 1)) def a(n): return b(n, 1) print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 10 2017
Formula
E.g.f.: cosh(x)*exp(x^2/(1-x)). More generally, e.g.f. for number of partitions of {1, 2, ...n} into lists with an even number of lists of size k is cosh(x^k)*exp(x/(1-x)-x^k).
E.g.f.: cosh(x)*exp(x^2/(1-x)) = 1/2*Q(0); Q(k) = 1+((2*x-1)^k)/(1-x/(x+((2*x-1)^k)*(k+1)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
a(n) ~ (exp(1)+exp(-1)) * 2^(-3/2) * exp(2*sqrt(n)-n-3/2) * n^(n-1/4) * (1 + (2/(1 + exp(2)) - 5/48)/sqrt(n)). - Vaclav Kotesovec, Jan 21 2017, extended Dec 01 2021
Extensions
More terms from David Wasserman, Feb 11 2009
a(0)=1 prepended by Alois P. Heinz, May 10 2016
Comments