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A288312 Number of endofunctions on [2n] such that the image size equals n.

%I A288312 #19 Jan 24 2022 16:53:46
%S A288312 1,2,84,10800,2857680,1285956000,880599202560,853262368358400,
%T A288312 1111400775560275200,1873276460474747328000,3967400888465895264384000,
%U A288312 10313998054713896966296473600,32291970618091110826769565696000,119851615755915509174015455948800000
%N A288312 Number of endofunctions on [2n] such that the image size equals n.
%H A288312 Alois P. Heinz, <a href="/A288312/b288312.txt">Table of n, a(n) for n = 0..195</a>
%F A288312 a(n) = Stirling2(2*n,n) * n! * binomial(2*n,n).
%F A288312 a(n) = A090657(2n,n) = A101817(2n,n) = A219859(2n,n).
%F A288312 a(n) ~ n^(2*n - 1/2) * 2^(4*n) / (sqrt(Pi*(1-c)) * c^n * (2-c)^n * exp(2*n)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - _Vaclav Kotesovec_, Jun 10 2017
%e A288312 a(1) = 2: (1,1), (2,2).
%p A288312 b:= proc(n, k) option remember; `if`(k=n, n!,
%p A288312       `if`(k=0, 0, n*(b(n-1, k-1)+b(n-1, k)*k/(n-k))))
%p A288312     end:
%p A288312 a:= n-> b(2*n, n):
%p A288312 seq(a(n), n=0..15);
%t A288312 Table[StirlingS2[2*n, n]*(2*n)!/n!, {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 10 2017 *)
%o A288312 (PARI) a(n)=stirling(2*n, n, 2)*n!*binomial(2*n, n); \\ _Indranil Ghosh_, Jul 04 2017
%o A288312 (Python)
%o A288312 from mpmath import *
%o A288312 mp.dps=100
%o A288312 def a(n): return int(stirling2(2*n, n)*fac(n)*binomial(2*n, n))
%o A288312 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Jul 04 2017
%Y A288312 Cf. A000142, A048993, A090657, A101817, A219859.
%K A288312 nonn
%O A288312 0,2
%A A288312 _Alois P. Heinz_, Jun 07 2017