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%I #20 Jul 22 2022 12:55:32
%S 0,1,10,63,336,1650,7722,35035,155584,680238,2939300,12584726,
%T 53488800,225990180,950094810,3977737875,16594533120,69018792150,
%U 286296636780,1184823735810,4893253404000,20171905282620,83020426503300
%N a(n) = (n-1)^2*binomial(2n,n)/(2*(n+1)).
%C a(n) = sum of valley abscissae in all Dyck paths of semilength n minus number of valleys in all Dyck paths of semilength (n). Example: a(3)=10; indeed, the Dyck paths of semilength 3, followed by their valley abscissae are UDUDUD (2,4), UDUUDD (2), UUDDUD (4), UUDUDD (3), UUUDDD ( ); therefore a(3)=2+4+2+4+3 - 5 = 10. Instead of Dyck paths one can consider Dyck words; then sum of valley abscissae corresponds to major index and number of valleys to number of descents.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see p. 236, Exercise 6.34 d.
%F a(n) = A002740(n+1) - A002054(n-1) (n >= 2).
%F a(n) = Sum_{k=0..(n-1)^2} k*A145884(n,k) for n >= 1.
%F a(n) = (n-1)^2*Cat(n)/2, where Cat(n)=binomial(2n,n)/(n+1)=A000108(n) are the Catalan numbers.
%F G.f.: 4*z^2*(8*z - 1 + 3*sqrt(1-4*z))/((1 + sqrt(1-4*z))^3*(1-4*z)^(3/2)).
%F D-finite with recurrence (n+1)*(n-2)^2*a(n) - 2*(2*n-1)*(n-1)^2*a(n-1) = 0. - _R. J. Mathar_, Aug 10 2017
%F E.g.f.: (exp(2*x) * ((1 + 2*x) * BesselI(0,2*x) - 2 * (2 - x) * BesselI(1,2*x)) - 1) / 2. - _Ilya Gutkovskiy_, Nov 03 2021
%p seq((1/2)*(n-1)^2*binomial(2*n,n)/(n+1),n=1..24);
%t Table[CatalanNumber[n]*(n - 1)^2/2, {n, 1, 23}] (* _Zerinvary Lajos_, Jul 08 2009 *)
%Y Cf. A000108, A002740, A002054, A145884.
%K nonn,easy
%O 1,3
%A _Emeric Deutsch_, Nov 06 2008
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