A232773 Permanent of the n X n matrix with numbers 1,2,...,n^2 in order across rows.

1, 1, 10, 450, 55456, 14480700, 6878394720, 5373548250000, 6427291156586496, 11157501095973529920, 26968983444160450560000, 87808164603589940623344000, 374818412822626584819196231680, 2050842983500342507649178541536000, 14112022767608502582976078751055052800

Offset: 0

Author

Franklin T. Adams-Watters, Nov 30 2013

Keywords

• nonn

Links

• Max Alekseyev, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A114533, A232788, A008277, A232818, A204248, A094638.

Programs

Maple

a:= n-> (-1)^n*add(n^k*Stirling1(n, n-k)*
        Stirling1(n+1, k+1)*(n-k)!*k!, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 02 2013

Mathematica

Table[(-1)^n * Sum[n^k * StirlingS1[n, n-k] * StirlingS1[n+1, k+1] * (n-k)! * k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec after Max Alekseyev, Nov 30 2013 *)

PARI

a(n) = (-1)^n * sum(k=0,n, n^k * stirling(n,n-k) * stirling(n+1,k+1) * (n-k)! * k! ) /* Max Alekseyev, Nov 30 2013 */

Python

from sympy.functions.combinatorial.numbers import stirling, factorial
def A232773(n): return abs(sum(n**k*stirling(n,n-k,kind=1,signed=True)*stirling(n+1,k+1,kind=1,signed=True)*factorial(n-k)*factorial(k) for k in range(n+1))) # Chai Wah Wu, Mar 25 2025

Formula

a(n) = (-1)^n * Sum_{k=0..n} n^k * Stirling1(n,n-k) * Stirling1(n+1,k+1) * (n-k)! * k!. - Max Alekseyev, Nov 30 2013

Limit_{n->oo} a(n)^(1/n)/n^3 = exp(-2). - Vaclav Kotesovec, Nov 30 2013

a(n) = A232788(n)*n!!, where n!! = A006882(n) is the double-factorial. - M. F. Hasler, Nov 30 2013

Extensions

More terms from W. Edwin Clark, Nov 30 2013

a(0)=1 prepended by Alois P. Heinz, Dec 02 2013