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A204264
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Permanent of the n-th principal submatrix of A051125.
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2
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1, 1, 6, 81, 2004, 78745, 4509474, 354647965, 36682768168, 4828756380525, 788293972077550, 156301383568936265, 36999739721382614268, 10307298862866331060865, 3338015288607791489356218, 1243533799932150850942393845, 528053459248835561744374233168
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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MAPLE
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with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, ()-> max(args)))):
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MATHEMATICA
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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 *)
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PROG
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(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]
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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