A232773 -id:A232773 - cp's OEIS frontend

A232788 A232773(n) / A006882(n): Permanent of the n X n matrix with elements [1,2,...,n^2], divided by n!!.

1, 1, 5, 150, 6932, 965380, 143299890, 51176650000, 16737737386944, 11806879466638656, 7023172771916784000, 8447153882019234307200, 8134080139379917205277696, 15176253254155788712392633600, 21875035292051870323313614135440, 59270306784445546617788929301760000

Offset: 0

M. F. Hasler, Nov 30 2013

nonn

Limit n->infinity a(n)^(1/n)/n^(5/2) = exp(-3/2). - Vaclav Kotesovec, Nov 08 2014

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A232773, A006882.

with(combinat):
a:= n-> (-1)^n *add(n^k *stirling1(n, n-k)*stirling1(n+1, k+1)
        *(n-k)!* k!, k=0..n)/doublefactorial(n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 02 2013

Flatten[{1,Table[(-1)^n*Sum[n^k*StirlingS1[n,n-k]*StirlingS1[n+1,k+1]*(n-k)!*k!,{k,0,n}]/n!!,{n,1,20}]}] (* Vaclav Kotesovec, Nov 08 2014 *)

n->(-1)^n*sum(k=0,n,n^k*stirling(n,n-k)*stirling(n+1,k+1)*(n-k)!*k!)/A006882(n)

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

A114533 Permanent of the n X n matrix with numbers prime(1),prime(2),...,prime(n^2) in order across rows.

Original entry on oeis.org

1, 2, 29, 3746, 1919534, 2514903732, 6571874957648, 30662862975835376, 228731722381012564816, 2641049525155781555257440, 43818773386947889568479502592, 1014966115357067575070490776083200, 31412851866841234377483875199638978304

Offset: 0

Simone Severini, Feb 15 2006

nonn

Previous name was : "a(n) = permanent of the n X n matrix M defined as follows: if we concatenate the rows of M to form a vector v of length n^2, v_i is the i-th prime number".

Vaclav Kotesovec, Table of n, a(n) for n = 0..36 (terms 0..20 from Alois P. Heinz)

Cf. A067276, A232773.

with(LinearAlgebra):
a:= n->`if`(n=0, 1, Permanent(Matrix(n, (i, j)->ithprime((i-1)*n+j)))):
seq(a(n), n=0..12);  # Alois P. Heinz, Dec 23 2013

a[n_] := Permanent[Table[Prime[(i-1)*n+j], {i, 1, n}, {j, 1, n}]]; a[0]=1; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 12}] (* Jean-François Alcover, Jan 07 2016, adapted from Maple *)
Join[{1},Table[Permanent[Partition[Prime[Range[n^2]],n]],{n,15}]] (* Harvey P. Dale, Aug 03 2019 *)

permRWN(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); n1=n-1; sg=1; m=1; nc=0; in=vector(n); x=in; for(i=1,n,x[i]=a[i,n]-sum(j=1,n,a[i,j])/2); p=prod(i=1,n,x[i]); while(m,sg=-sg; j=1; if((nc%2)!=0,j++; while(in[j-1]==0,j++)); in[j]=1-in[j]; z=2*in[j]-1; nc+=z; m=nc!=in[n1]; for(i=1,n,x[i]+=z*a[i,j]); p+=sg*prod(i=1,n,x[i])); return(2*(2*(n%2)-1)*p)
for(n=1,19,a=matrix(n,n,i,j,prime((i-1)*n+j)); print1(permRWN(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 11 2007

permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p)
for(n=1,23,a=matrix(n,n,i,j,prime((i-1)*n+j));print1(permRWNb(a)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 15 2007

{a(n) = matpermanent(matrix(n, n, i, j, prime((i-1)*n+j)))}
for(n=0, 25, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 13 2021

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 11 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 15 2007
New name from Michel Marcus, Nov 30 2013
a(0) inserted and a(12) by Alois P. Heinz, Dec 23 2013

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

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jan 14 2012

nonn

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

			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)

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

Cf. A002024, A232773, A232818, A094638.

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 *)

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 */

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

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

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

More terms from Max Alekseyev, Dec 02 2013

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

A232818 Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; nx+1, nx+2, ..., nx+n; ...; (n-1)n*x+1, (n-1)nx+2, ...,(n-1)nx+n].

Original entry on oeis.org

1, 6, 4, 216, 198, 36, 23040, 24640, 7200, 576, 5400000, 6375000, 2362500, 328800, 14400, 2351462400, 2982873600, 1285956000, 238533120, 19051200, 518400, 1707698764800, 2291162509440, 1100516981760, 245735819280, 27025656000, 1383117120, 25401600

Offset: 1

Vladimir Shevelev, Nov 30 2013

The degree of n-th polynomial is n-1.

Its leading coefficient is T(n,1) = n^n*(n-1)!^2*(n+1)/2. - M. F. Hasler, Dec 01 2013

			                                   1
                           6*x +   4
              216*x^2 +  198*x +  36
23040*x^3 + 24640*x^2 + 7200*x + 576
              ......

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A232773, A204248, A094638.

p[n_,x_]:=(-1)^n Sum[n^k x^k StirlingS1[n,n-k]StirlingS1[n+1,k+1](n-k)!k!,{k,0,n-1}];Flatten[Table[Reverse[CoefficientList[p[n,x],x]],{n,8}]] (* Peter J. C. Moses, Nov 30 2013 *)

P(n)=(-1)^n*sum(k=0,n-1,n^k*x^k*stirling(n,n-k)*stirling(n+1,k+1)*(n-k)!*k!)
apply(t->Vec(t),vector(7,n,P(n))) /* M. F. Hasler, Dec 01 2013 */

P_n(x) = (-1)^n * Sum_{k=0..n-1} c_k(n) * x^k, where c_k(n)= n^k * Stirling1(n,n-k) * Stirling1(n+1,k+1) * (n-k)! * k!.

P_n(1) = A232773; P_n(0) = n!^2, P_n(1/n) = A204248(n) is permanent of n X n Toeplitz matrix with the first row n,n-1,...,1 (see our comment in A204248).

More terms from Peter J. C. Moses, Nov 30 2013

A381723 a(n) = pos(M(n)), where M(n) is the n X n matrix with numbers 1, 2, ..., n^2 in order across rows, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 4, 225, 27728, 7240350, 3439197360, 2686774125000, 3213645578293248, 5578750547986764960, 13484491722080225280000, 43904082301794970311672000, 187409206411313292409598115840, 1025421491750171253824589270768000, 7056011383804251291488039375527526400

Offset: 1

Clark Kimberling, Mar 09 2025

nonn

			M(3) is the matrix with rows (1,2,3), (4,5,6), (7,8,9), determinant 0, permanent 450, negative part -225, and positive part 225.

Cf. A232773, A380661, A380662.

r[m_, n_] := Range[(m - 1) n + 1, m  n];
d = Table[Det[Table[r[m, n], {m, 1, n}]], {n, 1, 15}]
p = Table[Permanent[Table[r[m, n], {m, 1, n}]], {n, 1, 15}]
neg = (d - p)/2
pos = (d + p)/2

from sympy.functions.combinatorial.numbers import stirling, factorial
def A381723(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)))>>1 if n>2 else 3*n-2 # Chai Wah Wu, Mar 25 2025

a(n) = A232773(n)/2 for n >= 3.

cp's OEIS Frontend

A232788 A232773(n) / A006882(n): Permanent of the n X n matrix with elements [1,2,...,n^2], divided by n!!.

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A114533 Permanent of the n X n matrix with numbers prime(1),prime(2),...,prime(n^2) in order across rows.

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

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A232818 Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; n*x+1, n*x+2, ..., n*x+n; ...; (n-1)*n*x+1, (n-1)*n*x+2, ...,(n-1)*n*x+n].

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A381723 a(n) = pos(M(n)), where M(n) is the n X n matrix with numbers 1, 2, ..., n^2 in order across rows, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

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Formula

A232818 Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; nx+1, nx+2, ..., nx+n; ...; (n-1)n*x+1, (n-1)nx+2, ...,(n-1)nx+n].