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A182542 Second diagonal of triangle in A145879.

%I A182542 #37 Oct 15 2024 23:16:50
%S A182542 1,8,46,232,1093,4944,21778,94184,401930,1698160,7119516,29666704,
%T A182542 123012781,508019104,2091005866,8582372584,35141476126,143595498544,
%U A182542 585720020356,2385430111024,9701814930466,39411044641888,159926316674356,648348726966672,2626193752638388
%N A182542 Second diagonal of triangle in A145879.
%C A182542 Sum of valley heights over all Dyck n-paths. - _David Scambler_, Oct 05 2012
%H A182542 Alois P. Heinz, <a href="/A182542/b182542.txt">Table of n, a(n) for n = 3..200</a>
%F A182542 G.f. appears to be (1-2*x-sqrt(1-4*x))^2/(4*x*(1-4*x)). - _Mark van Hoeij_, Apr 19 2013
%F A182542 a(n) ~ 2^(2*n-2) * (1-4/(sqrt(Pi*n))). - _Vaclav Kotesovec_, Aug 13 2013
%F A182542 a(n) = 2*Sum_{i=0..n-1} 4^i*C(2*(n-i),n-i-2)/(n-i). - _Vladimir Kruchinin_, Mar 29 2019
%F A182542 a(n) = 4^(n-1) - C(2*n,n)*n/(n+1). - _Vladimir Kruchinin_, Jun 08 2020
%e A182542 Dyck 4-paths with nonzero valley heights are: UUUD(2)UDDD, UUUDD(1)UDD, UUD(1)UUDDD, UUD(1)UD(1)UDD, UUD(1)UDD(0)UD, and UD(0)UUD(1)UDD, with valley heights shown in parentheses, giving a(4) = 8. - _David Scambler_, Oct 05 2012
%t A182542 a[n_] := 4^(n - 1) - n CatalanNumber[n];
%t A182542 Array[a, 25, 3] (* _Peter Luschny_, Jun 08 2020 *)
%o A182542 (Maxima)
%o A182542 a(n):=2*sum((4^i*binomial(2*(n-i),n-i-2))/(n-i),i,0,n-1); /* _Vladimir Kruchinin_, Mar 29 2019  */
%Y A182542 Cf. A145879.
%K A182542 nonn
%O A182542 3,2
%A A182542 _N. J. A. Sloane_, May 04 2012
%E A182542 More terms from _Alois P. Heinz_, May 30 2012