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

%I A232773 #50 Mar 26 2025 16:34:02
%S A232773 1,1,10,450,55456,14480700,6878394720,5373548250000,6427291156586496,
%T A232773 11157501095973529920,26968983444160450560000,
%U A232773 87808164603589940623344000,374818412822626584819196231680,2050842983500342507649178541536000,14112022767608502582976078751055052800
%N A232773 Permanent of the n X n matrix with numbers 1,2,...,n^2 in order across rows.
%H A232773 Max Alekseyev, <a href="/A232773/b232773.txt">Table of n, a(n) for n = 0..100</a>
%F A232773 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
%F A232773 Limit_{n->oo} a(n)^(1/n)/n^3 = exp(-2). - _Vaclav Kotesovec_, Nov 30 2013
%F A232773 a(n) = A232788(n)*n!!, where n!! = A006882(n) is the double-factorial. - _M. F. Hasler_, Nov 30 2013
%p A232773 a:= n-> (-1)^n*add(n^k*Stirling1(n, n-k)*
%p A232773         Stirling1(n+1, k+1)*(n-k)!*k!, k=0..n):
%p A232773 seq(a(n), n=0..20);  # _Alois P. Heinz_, Dec 02 2013
%t A232773 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 *)
%o A232773 (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 */
%o A232773 (Python)
%o A232773 from sympy.functions.combinatorial.numbers import stirling, factorial
%o A232773 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
%Y A232773 Cf. A114533, A232788, A008277, A232818, A204248, A094638.
%K A232773 nonn
%O A232773 0,3
%A A232773 _Franklin T. Adams-Watters_, Nov 30 2013
%E A232773 More terms from _W. Edwin Clark_, Nov 30 2013
%E A232773 a(0)=1 prepended by _Alois P. Heinz_, Dec 02 2013