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%I #27 May 16 2022 04:53:07
%S 1,5,19,67,232,804,2806,9878,35072,125512,452388,1641028,5986993,
%T 21954973,80884423,299233543,1111219333,4140813373,15478839553,
%U 58028869153,218123355523,821908275547,3104046382351,11747506651599
%N Sum of the lengths of the second ascents in all Dyck paths of semilength n+2.
%C Also number of Dyck paths of semilength n+4 having length of second ascent equal to three. Example: a(1)=5 because we have UD(UUU)DUDDD, UD(UUU)DDUDD, UD(UUU)DDDUD, UUD(UUU)DDDD and UUDD(UUU)DDD (second ascents shown between parentheses). Partial sums of A002057. Column 3 of A114276. a(n)=absolute value of A104496(n+3).
%C Also number of Dyck paths of semilength n+3 that do not start with a pyramid (a pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis; here U=(1,1) and D=(1,-1); this definition differs from the one in A091866). Equivalently, a(n)=A127156(n+3,0). Example: a(1)=5 because we have UUDUDDUD, UUDUDUDD, UUUDUDDD, UUDUUDDD and UUUDDUDD. - _Emeric Deutsch_, Feb 27 2007
%H Vincenzo Librandi, <a href="/A114277/b114277.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = 4*Sum_{j=0..n} binomial(2*j+3, j)/(j+4).
%F G.f.: C^4/(1-z), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
%F a(n) = c(n+3) - (c(0) + c(1) + ... + c(n+2)), where c(k)=binomial(2k,k)/(k+1) is a Catalan number (A000108). - _Emeric Deutsch_, Feb 27 2007
%F D-finite with recurrence: n*(n+4)*a(n) = (5*n^2 + 14*n + 6)*a(n-1) - 2*(n+1)*(2*n+3)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 2^(2*n+7)/(3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) = exp((2*i*Pi)/3)-4*binomial(2*n+5,n+1)*hypergeom([1,3+n,n+7/2],[n+2,n+6],4)/ (n+5). - _Peter Luschny_, Feb 26 2017
%F a(n-1) = Sum_{i+j+k+l<n} C(i)C(j)C(k)C(l), where C=A000108 Catalan number. - _Yuchun Ji_, Jan 10 2019
%e a(3)=5 because the total length of the second ascents in UD(U)DUD, UD(UU)DD, UUDD(U)D, UUD(U)DD and UUUDDD (shown between parentheses) is 5.
%p a:=n->4*sum(binomial(2*j+3,j)/(j+4),j=0..n): seq(a(n),n=0..28);
%t Table[4*Sum[Binomial[2j+3,j]/(j+4),{j,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o (Python)
%o from functools import cache
%o @cache
%o def B(n, k):
%o if n <= 0 or k <= 0: return 0
%o if n == k: return 1
%o return B(n - 1, k) + B(n, k - 1)
%o def A114277(n): return B(n + 5, n + 1)
%o print([A114277(n) for n in range(24)]) # _Peter Luschny_, May 16 2022
%Y Cf. A002057, A114276, A104496, A127156, A279557.
%Y Cf. A014137 (n=1), A014138 (n=2), A001453 (n=3), this sequence (n=4), A143955 (n=5), A323224 (array).
%K nonn
%O 0,2
%A _Emeric Deutsch_, Nov 20 2005
%E More terms from _Emeric Deutsch_, Feb 27 2007
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