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 A204248 Permanent of the n-th principal submatrix of A002024. 4
 1, 7, 126, 4276, 234300, 18877020, 2100159600, 308417610816, 57786899446080, 13452134426136000, 3808606484711952000, 1288711254432792833280, 513583129024901529834240, 238093035025913233419052800, 127039392937347095305900800000, 77298350216325487808699492352000, 53201355385962541032225805510656000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS G. C. Greubel, Table of n, a(n) for n = 1..233 FORMULA a(n) = (-1)^n * Sum_{k=0..n-1} 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 EXAMPLE a(3) = per  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. -Vladimir Shevelev, Dec 01 2013 MATHEMATICA 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 *) Permanent[m_] :=   With[{a = Array[x, Length[m]]},    Coefficient[Times @@ (m.a), Times @@ a]]; Table[Permanent[m[n]], {n, 1, 15}]  (* A204248 *) PROG (PARI) 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 */ CROSSREFS Cf. A002024, A232773, A232818, A094638. Sequence in context: A180583 A295412 A202798 * A084940 A246648 A139987 Adjacent sequences:  A204245 A204246 A204247 * A204249 A204250 A204251 KEYWORD nonn AUTHOR Clark Kimberling, Jan 14 2012 EXTENSIONS More terms from Max Alekseyev, Dec 02 2013 STATUS approved

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Last modified March 19 13:25 EDT 2019. Contains 321330 sequences. (Running on oeis4.)