cp's OEIS Frontend

A119358 Number of n-element subsets of [2n] having an even sum.

%I A119358 #29 Mar 30 2023 16:08:51
%S A119358 1,1,2,10,38,126,452,1716,6470,24310,92252,352716,1352540,5200300,
%T A119358 20056584,77558760,300546630,1166803110,4537543340,17672631900,
%U A119358 68923356788,269128937220,1052049129144,4116715363800,16123803193628,63205303218876,247959261273752
%N A119358 Number of n-element subsets of [2n] having an even sum.
%C A119358 Old name was: Central coefficients of number triangle A119326.
%H A119358 Alois P. Heinz, <a href="/A119358/b119358.txt">Table of n, a(n) for n = 0..1000</a>
%F A119358 G.f.: (1/sqrt(1-4x)+1/sqrt(1+4x^2))/2.
%F A119358 a(n) = Sum_{k=0..floor(n/2)} C(n,2k)^2.
%F A119358 a(n) = C(2n,n)/2+sin(Pi*(n+1)/2)*C(n,n/2)/2.
%F A119358 a(n) = A119326(2n,n).
%F A119358 a(n) = A071688(n) + A119359(n) for n>=1.
%F A119358 D-finite with recurrence n*(n-1)*(10*n-29)*a(n) +2*(n-1)*(5*n^2-74*n+164)*a(n-1) +4*(-40*n^3+310*n^2 -744*n+559)*a(n-2) +8*(n-2)*(5*n^2-74*n+164)*a(n-3) -16*(25*n-42)*(n-3)*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Nov 05 2012
%F A119358 a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], 1). - _Vladimir Reshetnikov_, Oct 04 2016
%F A119358 a(n) = A282011(2n,n). - _Alois P. Heinz_, Feb 04 2017
%e A119358 a(3) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}. - _Alois P. Heinz_, Feb 04 2017
%p A119358 a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
%p A119358      ((4*n-10)*(5*n^2-10*n+4)*(a(n-1)+4*(n-2)*a(n-3)
%p A119358       /(n-1))/(5*n^2-20*n+19)-4*(n-1)*a(n-2))/n)
%p A119358     end:
%p A119358 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 26 2018
%t A119358 Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, 1], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 04 2016 *)
%Y A119358 Cf. A071688, A119326, A119359, A169888, A282011.
%Y A119358 Column k=2 of A318557.
%K A119358 easy,nonn
%O A119358 0,3
%A A119358 _Paul Barry_, May 16 2006
%E A119358 New name from _Alois P. Heinz_, Feb 04 2017