A204264 Permanent of the n-th principal submatrix of A051125.
1, 1, 6, 81, 2004, 78745, 4509474, 354647965, 36682768168, 4828756380525, 788293972077550, 156301383568936265, 36999739721382614268, 10307298862866331060865, 3338015288607791489356218, 1243533799932150850942393845, 528053459248835561744374233168, 253548768266772102234162057089429
Offset: 0
Keywords
Links
- Discussion at dxdy.ru, Permanent of a matrix, (in Russian) (2023).
Programs
-
Maple
with(LinearAlgebra): a:= n-> `if`(n=0, 1, Permanent(Matrix(n, ()-> max(args)))): seq(a(n), n=0..16); # Alois P. Heinz, Nov 14 2016
-
Mathematica
f[i_, j_] := Max[i, j]; 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}]] (* A051125 *) Permanent[m_] := With[{a = Array[x, Length[m]]}, Coefficient[Times @@ (m.a), Times @@ a]]; Table[Permanent[m[n]], {n, 1, 12}] (* A204264 *) -
PARI
a(n)=if(n==0, 1, my(v1, x='x); v1=vector(n+1, i, i--; i!*x^i); for(i=1, n, for(j=i, n, my(A=intformal((j-i)^2*v1[j])); v1[j+1] = A + subst(v1[j+1] - A, x, n - i + 1))); v1[n+1]) \\ Mikhail Kurkov, Aug 03 2023 [verification needed]
Formula
a(n) = f(n, n, n, 0) for n >= 0 where f(n, q, m, x) = g(n, q, m, x) + f(n, q-1, m, m - q + 1) - g(n, q, m, m - q + 1) for n >= 0, q > 0, m >= 0 with f(n, 0, m, x) = n!*x^n for n >= 0, m >= 0 and where g(n, q, m, x) = Integral (n-q)^2*f(n-1, q, m, x) dx for n > 0, q > 0, m > 0 (formula due to user with the nickname Null on a scientific forum dxdy.ru). - Mikhail Kurkov, Aug 03 2023 [verification needed]
a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.2311615029480675454906706378647589... and c = 6.0549338839461696639515756882139886... - Vaclav Kotesovec, Aug 05 2023
Extensions
a(0)=1 prepended and more terms added by Alois P. Heinz, Nov 14 2016