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A204248 Permanent of the n-th principal submatrix of A002024.

%I A204248 #62 Jun 09 2025 00:58:24
%S A204248 1,1,7,126,4276,234300,18877020,2100159600,308417610816,
%T A204248 57786899446080,13452134426136000,3808606484711952000,
%U A204248 1288711254432792833280,513583129024901529834240,238093035025913233419052800,127039392937347095305900800000,77298350216325487808699492352000
%N A204248 Permanent of the n-th principal submatrix of A002024.
%C A204248 a(n) is permanent of Toeplitz matrix
%C A204248   n    n-1  n-2   ...  3   2  1
%C A204248   n+1  n    n-1   ...  4   3  2
%C A204248   n+2  n+1  n     ...  5   4  3
%C A204248                 .......
%C A204248   2n-1 2n-2 2n-3  ... n+2 n+1 n. - _Vladimir Shevelev_, Dec 01 2013
%H A204248 G. C. Greubel, <a href="/A204248/b204248.txt">Table of n, a(n) for n = 0..233</a>
%F A204248 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,n-k) * Stirling1(n+1,k+1) * (n-k)! * k!. - _Vladimir Shevelev_, Dec 01 2013
%F A204248 Limit n->infinity a(n)^(1/n)/n^2 = -2*c^2/(exp(2)*(1+2*c)) = 0.33230326707622..., where c = LambertW(-1,-1/(2*exp(1/2))) = -1.756431208626... - _Vaclav Kotesovec_, Dec 10 2013
%F A204248 a(n) ~ 2.531082868731093... * (-2*c^2/(exp(2)*(1+2*c)))^n * n^(2*n+1/2), where c = LambertW(-1,-1/(2*exp(1/2))). - _Vaclav Kotesovec_, Dec 10 2013
%F A204248 a(n) = n!*abs(Stirling1(2*n+1,n+1))/C(2*n+1,n). - _Chai Wah Wu_, Jun 08 2025
%e A204248 From _Vladimir Shevelev_, Dec 01 2013: (Start)
%e A204248 a(3) = permanent ( 3 2 1 ) = 3*17 + 2*22 + 1*31 = 126.
%e A204248                  ( 4 3 2 )
%e A204248                  ( 5 4 3 )
%e A204248 and
%e A204248 a(3) = |stirling1(3,3)*stirling1(4,1)|*6*1 + |stirling1(3,2)*stirling1(4,2)|*2*1 + |stirling1(3,1)*stirling1(4,3)|*1*2 = 1*6*6*1 + 3*11*2*1 + 2*6*1*2 = 126. (End)
%t A204248 f[i_, j_] := i + j - 1;
%t A204248 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204248 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204248 Flatten[Table[f[i, n + 1 - i],
%t A204248   {n, 1, 12}, {i, 1, n}]]  (* A002024 *)
%t A204248 Join[{1},Table[Permanent[m[n]], {n, 1, 15}]]  (* A204248 *)
%o A204248 (PARI) a(n) = (-1)^n * sum(k=0, n-1, stirling(n, n-k) * stirling(n+1, k+1) * (n-k)! * k! ) /* _Max Alekseyev_, Dec 02 2013 */
%o A204248 (Python)
%o A204248 from math import factorial
%o A204248 from sympy.functions.combinatorial.numbers import stirling
%o A204248 def A204248(n): return sum(stirling(n,n-k,kind=1)*stirling(n+1,k+1,kind=1)*factorial(n-k)*factorial(k) for k in range(n)) if n else 1 # _Chai Wah Wu_, Oct 16 2022
%o A204248 (Python)
%o A204248 from math import factorial, comb
%o A204248 from sympy.functions.combinatorial.numbers import stirling
%o A204248 def A204248(n): return factorial(n)*stirling(m:=(n<<1)+1,n+1,kind=1)//comb(m,n) # _Chai Wah Wu_, Jun 08 2025
%Y A204248 Cf. A002024, A232773, A232818, A094638.
%K A204248 nonn
%O A204248 0,3
%A A204248 _Clark Kimberling_, Jan 14 2012
%E A204248 More terms from _Max Alekseyev_, Dec 02 2013
%E A204248 a(0)=1 prepended by _Pontus von Brömssen_, Jan 30 2021