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

Author

N. J. A. Sloane

Keywords

• nonn

References

• 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).

Links

• 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]

Crossrefs

Cf. A000085, A081919, A002772.

Programs

Maple

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

Mathematica

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 *)

Sage

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

Formula

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

Extensions

More terms from Sean A. Irvine, Aug 18 2014

Expanded definition from Peter Luschny, Aug 21 2014