cp's OEIS Frontend

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

%I A002771 M1269 N0488 #50 Oct 20 2023 11:54:30
%S A002771 1,2,4,13,41,226,1072,9374,60958,723916,5892536,86402812,837641884,
%T A002771 14512333928,162925851376,3252104882056,41477207604872,
%U A002771 937014810365584,13380460644770848,337457467862898896,5333575373478669136,148532521250931168352
%N A002771 Number of terms in a skew determinant: a(n) = (A000085(n) + A081919(n))/2.
%D A002771 T. Muir, The expression of any bordered skew determinant as a sum of products of Pfaffians, Proc. Roy. Soc. Edinburgh, 21 (1896), 342-359.
%D A002771 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002771 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002771 Alois P. Heinz, <a href="/A002771/b002771.txt">Table of n, a(n) for n = 1..400</a>
%H A002771 T. Muir, <a href="/A002772/a002772.pdf">The expression of any bordered skew determinant as a sum of products of Pfaffians</a>, Proc. Roy. Soc. Edinburgh, 21 (1896), 342-359. [Annotated scan of pages 354-357 only]
%H A002771 T. Muir, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM9350.0003.001">The Theory of Determinants in the Historical Order of Development</a>, 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
%H A002771 T. Muir, <a href="/A002135/a002135_1.pdf">The Theory of Determinants in the Historical Order of Development</a>, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]
%F A002771 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
%F A002771 (-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
%F A002771 a(n) = (hyper2F0([-n/2,(1-n)/2],[],2)+hyper3F0([1/2,-n/2,(1-n)/2],[],4))/2. - _Peter Luschny_, Aug 21 2014
%F A002771 a(n) ~ ((-1)^n*exp(-1) + exp(1)) * n^n / (2*exp(n)). - _Vaclav Kotesovec_, Sep 12 2014
%p A002771 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
%p A002771 # second Maple program:
%p A002771 a:= proc(n) a(n):= `if`(n<5, [1$2, 2, 4, 13][n+1],
%p A002771      ((2*n-5) *a(n-1) +(n-1)*(n^2-4*n+1) *a(n-2)
%p A002771       -(2*n-5)*(n-1)*(n-2) *a(n-3))/(n-4)
%p A002771       +(n-1)*(n-2)*(n-3) *(a(n-5)-a(n-4)))
%p A002771     end:
%p A002771 seq(a(n), n=1..25);  # _Alois P. Heinz_, Aug 18 2014
%t A002771 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_ *)
%o A002771 (Sage)
%o A002771 def A002771(n):
%o A002771     A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
%o A002771     A081919 = lambda n: hypergeometric([1/2,-n/2,(1-n)/2], [], 4)
%o A002771     return ((A000085(n) + A081919(n))/2).n()
%o A002771 [round(A002771(n)) for n in (1..22)]  # _Peter Luschny_, Aug 21 2014
%Y A002771 Cf. A000085, A081919, A002772.
%K A002771 nonn
%O A002771 1,2
%A A002771 _N. J. A. Sloane_
%E A002771 More terms from _Sean A. Irvine_, Aug 18 2014
%E A002771 Expanded definition from _Peter Luschny_, Aug 21 2014