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A002771 Number of terms in a skew determinant: a(n) = (A000085(n) + A081919(n))/2.
(Formerly M1269 N0488)
4

%I M1269 N0488 #50 Oct 20 2023 11:54:30

%S 1,2,4,13,41,226,1072,9374,60958,723916,5892536,86402812,837641884,

%T 14512333928,162925851376,3252104882056,41477207604872,

%U 937014810365584,13380460644770848,337457467862898896,5333575373478669136,148532521250931168352

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

%D 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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A002771/b002771.txt">Table of n, a(n) for n = 1..400</a>

%H 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 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 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 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 (-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 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 a(n) ~ ((-1)^n*exp(-1) + exp(1)) * n^n / (2*exp(n)). - _Vaclav Kotesovec_, Sep 12 2014

%p 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 # second Maple program:

%p a:= proc(n) a(n):= `if`(n<5, [1$2, 2, 4, 13][n+1],

%p ((2*n-5) *a(n-1) +(n-1)*(n^2-4*n+1) *a(n-2)

%p -(2*n-5)*(n-1)*(n-2) *a(n-3))/(n-4)

%p +(n-1)*(n-2)*(n-3) *(a(n-5)-a(n-4)))

%p end:

%p seq(a(n), n=1..25); # _Alois P. Heinz_, Aug 18 2014

%t 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 (Sage)

%o def A002771(n):

%o A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)

%o A081919 = lambda n: hypergeometric([1/2,-n/2,(1-n)/2], [], 4)

%o return ((A000085(n) + A081919(n))/2).n()

%o [round(A002771(n)) for n in (1..22)] # _Peter Luschny_, Aug 21 2014

%Y Cf. A000085, A081919, A002772.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, Aug 18 2014

%E Expanded definition from _Peter Luschny_, Aug 21 2014

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)