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A288312 Number of endofunctions on [2n] such that the image size equals n. 3
1, 2, 84, 10800, 2857680, 1285956000, 880599202560, 853262368358400, 1111400775560275200, 1873276460474747328000, 3967400888465895264384000, 10313998054713896966296473600, 32291970618091110826769565696000, 119851615755915509174015455948800000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..195

FORMULA

a(n) = Stirling2(2*n,n) * n! * binomial(2*n,n).

a(n) = A090657(2n,n) = A101817(2n,n).

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

EXAMPLE

a(1) = 2: (1,1), (2,2).

MAPLE

b:= proc(n, k) option remember; `if`(k=n, n!,

      `if`(k=0, 0, n*(b(n-1, k-1)+b(n-1, k)*k/(n-k))))

    end:

a:= n-> b(2*n, n):

seq(a(n), n=0..15);

MATHEMATICA

Table[StirlingS2[2*n, n]*(2*n)!/n!, {n, 0, 20}] (* Vaclav Kotesovec, Jun 10 2017 *)

PROG

(PARI) a(n)=stirling(2*n, n, 2)*n!*binomial(2*n, n); \\ Indranil Ghosh, Jul 04 2017

(Python)

from mpmath import *

mp.dps=100

def a(n): return int(stirling2(2*n, n)*fac(n)*binomial(2*n, n))

print [a(n) for n in range(21)] # Indranil Ghosh, Jul 04 2017

CROSSREFS

Cf. A000142, A048993, A090657, A101817.

Sequence in context: A215263 A265591 A157063 * A289198 A318128 A181119

Adjacent sequences:  A288309 A288310 A288311 * A288313 A288314 A288315

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 07 2017

STATUS

approved

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Last modified February 25 22:55 EST 2020. Contains 332270 sequences. (Running on oeis4.)