A002772 -id:A002772 - cp's OEIS frontend

A002771 Number of terms in a skew determinant: a(n) = (A000085(n) + A081919(n))/2.

1, 2, 4, 13, 41, 226, 1072, 9374, 60958, 723916, 5892536, 86402812, 837641884, 14512333928, 162925851376, 3252104882056, 41477207604872, 937014810365584, 13380460644770848, 337457467862898896, 5333575373478669136, 148532521250931168352

Offset: 1

N. J. A. Sloane

nonn

T. Muir, The expression of any bordered skew determinant as a sum of products of Pfaffians, Proc. Roy. Soc. Edinburgh, 21 (1896), 342-359.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..400
T. Muir, The expression of any bordered skew determinant as a sum of products of Pfaffians, Proc. Roy. Soc. Edinburgh, 21 (1896), 342-359. [Annotated scan of pages 354-357 only]
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]

Cf. A000085, A081919, A002772.

seq(sum(binomial(n, 2*k) * doublefactorial(2*k-1) * (1+doublefactorial(2*k-1))/2, k=0..floor(n/2)), n=1..40); # Sean A. Irvine, Aug 18 2014
# second Maple program:
a:= proc(n) a(n):= `if`(n<5, [1$2, 2, 4, 13][n+1],
     ((2*n-5) *a(n-1) +(n-1)*(n^2-4*n+1) *a(n-2)
      -(2*n-5)*(n-1)*(n-2) *a(n-3))/(n-4)
      +(n-1)*(n-2)*(n-3) *(a(n-5)-a(n-4)))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Aug 18 2014

a[n_] := Sum[Binomial[n, 2*k] * (2*k-1)!! * (1 + (2*k-1)!!) / 2, {k, 0, n/2}]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 26 2015, after Sean A. Irvine *)

def A002771(n):
    A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
    A081919 = lambda n: hypergeometric([1/2,-n/2,(1-n)/2], [], 4)
    return ((A000085(n) + A081919(n))/2).n()
[round(A002771(n)) for n in (1..22)]  # Peter Luschny, Aug 21 2014

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * (2*k-1)!! * (1 + (2*k-1)!!) / 2. - Sean A. Irvine, Aug 18 2014
(-n+4)*a(n) +(2*n-5)*a(n-1) +(n-1)*(n^2-4*n+1)*a(n-2) -(2*n-5)*(n-1)*(n-2)*a(n-3) -(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) +(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 19 2014
a(n) = (hyper2F0([-n/2,(1-n)/2],[],2)+hyper3F0([1/2,-n/2,(1-n)/2],[],4))/2. - Peter Luschny, Aug 21 2014
a(n) ~ ((-1)^n*exp(-1) + exp(1)) * n^n / (2*exp(n)). - Vaclav Kotesovec, Sep 12 2014

More terms from Sean A. Irvine, Aug 18 2014

Expanded definition from Peter Luschny, Aug 21 2014

A243107 Number of terms in a bordered skew determinant.

Original entry on oeis.org

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

Sean A. Irvine, Aug 19 2014

nonn

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

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.

Cf. A002771, A002772.

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

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 *)

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

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

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

cp's OEIS Frontend

A002771 Number of terms in a skew determinant: a(n) = (A000085(n) + A081919(n))/2.

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