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

1, 1, 7, 126, 4276, 234300, 18877020, 2100159600, 308417610816, 57786899446080, 13452134426136000, 3808606484711952000, 1288711254432792833280, 513583129024901529834240, 238093035025913233419052800, 127039392937347095305900800000, 77298350216325487808699492352000

Offset: 0

Author

Clark Kimberling, Jan 14 2012

Keywords

• nonn

Comments

a(n) is permanent of Toeplitz matrix

n n-1 n-2 ... 3 2 1

n+1 n n-1 ... 4 3 2

n+2 n+1 n ... 5 4 3

.......

2n-1 2n-2 2n-3 ... n+2 n+1 n. - Vladimir Shevelev, Dec 01 2013

Examples

			From _Vladimir Shevelev_, Dec 01 2013: (Start)
a(3) = permanent ( 3 2 1 ) = 3*17 + 2*22 + 1*31 = 126.
                 ( 4 3 2 )
                 ( 5 4 3 )
and
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)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..233

Crossrefs

Cf. A002024, A232773, A232818, A094638.

Programs

Mathematica

f[i_, j_] := i + j - 1;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]  (* A002024 *)
Join[{1},Table[Permanent[m[n]], {n, 1, 15}]]  (* 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 */

Python

from math import factorial
from sympy.functions.combinatorial.numbers import stirling
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

Python

from math import factorial, comb
from sympy.functions.combinatorial.numbers import stirling
def A204248(n): return factorial(n)*stirling(m:=(n<<1)+1,n+1,kind=1)//comb(m,n) # Chai Wah Wu, Jun 08 2025

Formula

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

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

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

a(n) = n!*abs(Stirling1(2*n+1,n+1))/C(2*n+1,n). - Chai Wah Wu, Jun 08 2025

Extensions

More terms from Max Alekseyev, Dec 02 2013

a(0)=1 prepended by Pontus von Brömssen, Jan 30 2021