A243107 Number of terms in a bordered skew determinant.

1, 1, 2, 4, 13, 41, 226, 1072, 9059, 58123, 657766, 5268836, 73980787, 707506879, 11823958238, 131277234376, 2542107619081, 32122718085497, 706963537444114, 10015472595953908, 246853433179370621, 3874536631479770761, 105709617658879558402

Offset

0,3

Comments

Possibly a different attempt to count the same bordered skew determinants as in A002772.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.

J. J. Sylvester, Note on determinants and duadic disynthemes, American J of Math, ii, (1879), 89-90, 214-222.

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n, n - 2*k) * A002370(k).

E.g.f.: exp(x+x^2/4) / (1-x^2)^(1/4).

a(n) ~ n! * GAMMA(3/4) * (exp(5/4) + (-1)^n * exp(-3/4)) / (Pi * 2^(3/4)* n^(3/4)). - Vaclav Kotesovec, Aug 20 2014

Maple

a:= proc(n) option remember; `if`(n<4, [1$2, 2, 4][n+1],
      (2*a(n-1)+2*(n-1)^2*a(n-2)-2*(n-1)*(n-2)*a(n-3)
       -(n-1)*(n-2)*(n-3)*a(n-4))/2)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 19 2014

Mathematica

b[n_] := Gamma[n+1/2] HypergeometricPFQ[{1/4, -n}, {}, -4]/Sqrt[Pi];
a[n_] := Sum[Binomial[n, n-2k] b[k], {k, 0, n/2}];
a /@ Range[0, 30]
(* Second program: *)
a[n_] := a[n] = If[n < 4, {1, 1, 2, 4}[[n+1]], (2a[n-1] + 2(n-1)^2 a[n-2] - 2(n-1)(n-2)a[n-3] - (n-1)(n-2)(n-3) a[n-4])/2];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)

Prog

(PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(x+x^2/4) / (1-x^2)^(1/4))) \\ Joerg Arndt, Aug 20 2014

Crossrefs

Cf. A002771, A002772.

Sequence in context: A355194 A087214 A259239 * A002771 A284159 A050624

Adjacent sequences: A243104 A243105 A243106 * A243108 A243109 A243110

Keyword

nonn

Author

Sean A. Irvine, Aug 19 2014

Extensions

a(0)=1 prepended by Joerg Arndt, Aug 24 2014

Status

approved