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%I #35 Sep 08 2022 08:44:53
%S 0,0,1,1,4,5,15,21,56,84,210,330,792,1287,3003,5005,11440,19448,43758,
%T 75582,167960,293930,646646,1144066,2496144,4457400,9657700,17383860,
%U 37442160,67863915,145422675,265182525,565722720,1037158320,2203961430,4059928950,8597496600,15905368710,33578000610,62359143990
%N a(n) = binomial(n, floor(n/2)-1).
%C Number of returns to the axis in all left factors of Dyck paths of length n. Example: a(4)=4 because in U(D)U(D), U(D)UU, UUD(D), UUDU, UUUD, and UUUU we have a total of 2+1+1+0+0+0=4 returns to the axis (shown between parentheses); here U=(1,1) and D=(1,-1). - _Emeric Deutsch_, Jun 06 2011
%C a(n) is the number of subsets of {1,2,...,n} that contain exactly 1 more even than odd elements. For example, a(6) = 15 and the 15 sets are {2}, {4}, {6}, {1,2,4}, {1,2,6}, {1,4,6}, {2,3,4}, {2,3,6}, {2,4,5}, {2,5,6}, {3,4,6}, {4,5,6}, {1,2,3,4,6}, {1,2,4,5,6}, {2,3,4,5,6}. - _Enrique Navarrete_, Dec 20 2019
%H Robert Israel, <a href="/A037955/b037955.txt">Table of n, a(n) for n = 0..3324</a>
%F E.g.f.: Bessel_I(2,2*x) + Bessel_I(3,2*x). - _Paul Barry_, Feb 28 2006
%F G.f.: g(z) = z^2*c^3/(1-z*c), where c = (1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2. - _Emeric Deutsch_, Jun 06 2011
%F (n+3)*(n-2)*a(n) +2*n*a(n-1) +4*n*(n-1)*a(n-2) = 0. - _R. J. Mathar_, Nov 30 2012
%F a(n) = binomial(n, (n-2)/2), n even; a(n) = binomial(n, (n-3)/2), n odd. - _Enrique Navarrete_, Dec 20 2019
%p seq(binomial(n, floor((n-2)/2)), n = 0..40);
%t Table[Binomial[n,Floor[n/2-1]], {n,0,40}] (* _Wesley Ivan Hurt_, Oct 16 2013 *)
%o (PARI) vector(41, n, binomial(n-1, (n-3)\2) ) \\ _G. C. Greubel_, Dec 31 2019
%o (Magma) [Binomial(n, Floor((n-2)/2)): n in [0..40]]; // _G. C. Greubel_, Dec 31 2019
%o (Sage) [binomial(n, floor(n/2)-1) for n in (0..40)] # _G. C. Greubel_, Dec 31 2019
%Y Cf. A037952, A047182.
%K nonn
%O 0,5
%A _N. J. A. Sloane_
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