0,2

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 (deutsch(AT)duke.poly.edu), Feb 27 2007

a(n)=4sum(binomial(2j+3, j)/(j+4), j=0..n). G.f.=C^4/(1-z), where C=[1-sqrt(1-4z)]/(2z) 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 (deutsch(AT)duke.poly.edu), Feb 27 2007

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.

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

Cf. A002057, A114276, A104496.

Cf. A127156.

Adjacent sequences: A114274 A114275 A114276 this_sequence A114278 A114279 A114280

Sequence in context: A067325 A121525 A035344 this_sequence A104496 A001435 A092492

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 20 2005

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2007