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%I #27 Apr 21 2024 22:21:57
%S 1,1,6,81,2004,78745,4509474,354647965,36682768168,4828756380525,
%T 788293972077550,156301383568936265,36999739721382614268,
%U 10307298862866331060865,3338015288607791489356218,1243533799932150850942393845,528053459248835561744374233168
%N Permanent of the n-th principal submatrix of A051125.
%H Discussion at dxdy.ru, <a href="https://dxdy.ru/topic154879.html">Permanent of a matrix</a>, (in Russian) (2023).
%F 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]
%F a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.2311615029480675454906706378647589... and c = 6.0549338839461696639515756882139886... - _Vaclav Kotesovec_, Aug 05 2023
%p with(LinearAlgebra):
%p a:= n-> `if`(n=0, 1, Permanent(Matrix(n, ()-> max(args)))):
%p seq(a(n), n=0..16); # _Alois P. Heinz_, Nov 14 2016
%t f[i_, j_] := Max[i, j];
%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t TableForm[m[8]] (* 8x8 principal submatrix *)
%t Flatten[Table[f[i, n + 1 - i],
%t {n, 1, 12}, {i, 1, n}]] (* A051125 *)
%t Permanent[m_] :=
%t With[{a = Array[x, Length[m]]},
%t Coefficient[Times @@ (m.a), Times @@ a]];
%t Table[Permanent[m[n]], {n, 1, 12}] (* A204264 *)
%o (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]
%Y Cf. A051125, A204262.
%K nonn,changed
%O 0,3
%A _Clark Kimberling_, Jan 14 2012
%E a(0)=1 prepended and more terms added by _Alois P. Heinz_, Nov 14 2016
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