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A114277 Sum of the lengths of the second ascents in all Dyck paths of semilength n+2. 8
1, 5, 19, 67, 232, 804, 2806, 9878, 35072, 125512, 452388, 1641028, 5986993, 21954973, 80884423, 299233543, 1111219333, 4140813373, 15478839553, 58028869153, 218123355523, 821908275547, 3104046382351, 11747506651599 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

FORMULA

a(n) = 4*Sum_{j=0..n} binomial(2*j+3, j)/(j+4).

G.f.: C^4/(1-z), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.

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

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

a(n) ~ 2^(2*n+7)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

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

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

EXAMPLE

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.

MAPLE

a:=n->4*sum(binomial(2*j+3, j)/(j+4), j=0..n): seq(a(n), n=0..28);

MATHEMATICA

Table[4*Sum[Binomial[2j+3, j]/(j+4), {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 19 2012 *)

CROSSREFS

Cf. A002057, A114276, A104496, A127156, A279557.

Sequence in context: A121525 A163872 A035344 * A104496 A001435 A092492

Adjacent sequences:  A114274 A114275 A114276 * A114278 A114279 A114280

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 20 2005

EXTENSIONS

More terms from Emeric Deutsch, Feb 27 2007

STATUS

approved

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Last modified June 4 10:43 EDT 2020. Contains 334825 sequences. (Running on oeis4.)