A002771 Number of terms in a skew determinant: a(n) = (A000085(n) + A081919(n))/2. (Formerly M1269 N0488)

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

Offset

1,2

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]

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

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 *)

Prog

(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

Crossrefs

Cf. A000085, A081919, A002772.

Sequence in context: A087214 A259239 A243107 * A284159 A050624 A135501

Adjacent sequences: A002768 A002769 A002770 * A002772 A002773 A002774

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Aug 18 2014

Expanded definition from Peter Luschny, Aug 21 2014

Status

approved