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A204262 Permanent of the n-th principal submatrix of A003983.

%I A204262 #49 Oct 29 2024 12:08:00
%S A204262 1,1,3,19,209,3545,85803,2807723,119377321,6397099105,421772316915,
%T A204262 33552418294339,3168847554832961,350514662908385321,
%U A204262 44885099167514403963,6587836555407268741019,1098597117953239632728089,206564512095561068049417265,43495029251774783469442768323
%N A204262 Permanent of the n-th principal submatrix of A003983.
%H A204262 Alois P. Heinz, <a href="/A204262/b204262.txt">Table of n, a(n) for n = 0..264</a>
%H A204262 Discussion at dxdy.ru, <a href="https://dxdy.ru/topic154879.html">Permanent of a matrix</a>, (in Russian) (2023).
%H A204262 Terence Tao, <a href="https://mathoverflow.net/a/450607/231922">Remarkable recursions for the A204262</a>, answer to question on MathOverflow (2023).
%F A204262 From _Mikhail Kurkov_, Aug 03 2023: (Start)
%F A204262 a(n) = f(n, n, 0) for n >= 0 where f(n, q, x) = g(n, q, x) + f(n, q-1, q) - g(n, q, q) for n >= 0, q > 0 with f(n, 0, x) = n!*x^n for n >= 0 and where g(n, q, x) = Integral (n-q)^2*f(n-1, q, x) dx for n > 0, q > 0 (formula due to user with the nickname Null on a scientific forum dxdy.ru).
%F A204262 a(n) = R(n-1, 0) for n > 0 with a(0) = 1 where R(n, q) = Sum_{j=0..q+1} binomial(q+1, j)*(j+1)*R(n-1, j) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0.
%F A204262 Both results were proved by _Terence Tao_, see Links section. (End)
%F A204262 Conjecture: Limit_{n->oo} (a(n)/n!^2)^(1/n) = 2/Pi. - _Vaclav Kotesovec_, Aug 05 2023
%p A204262 with(LinearAlgebra):
%p A204262 a:= n-> `if`(n=0, 1, Permanent(Matrix(n, ()-> min(args)))):
%p A204262 seq(a(n), n=0..16);  # _Alois P. Heinz_, Nov 14 2016
%t A204262 f[i_, j_] := Min[i, j];
%t A204262 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204262 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204262 Flatten[Table[f[i, n + 1 - i],
%t A204262   {n, 1, 12}, {i, 1, n}]]      (* A003983 *)
%t A204262 Permanent[m_] :=
%t A204262   With[{a = Array[x, Length[m]]},
%t A204262    Coefficient[Times @@ (m.a), Times @@ a]];
%t A204262 Table[Permanent[m[n]], {n, 1, 15}]    (* A204262 *)
%o A204262 (PARI) a(n)={my(S,z,v=vector(n));for(i=0,n!-1,v=numtoperm(n,i);z=1;for(j=1,n,z*= n+1-max(j,v[j]));S+=z);return(S)} \\ _R. J. Cano_, Nov 14 2016
%o A204262 (PARI) upto(n)=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, i))); v1 \\ _Mikhail Kurkov_, Aug 03 2023
%Y A204262 Cf. A003983, A204264.
%K A204262 nonn
%O A204262 0,3
%A A204262 _Clark Kimberling_, Jan 14 2012
%E A204262 a(0)=1 prepended and more terms added by _Alois P. Heinz_, Nov 14 2016