2,2

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=2. Example: For n=3 there are 6 paths EEENNN, EENENN, EENNEN, EENNNE, ENEENN and NEEENN. - Herbert Kociemba (kociemba(AT)t-online.de), May 23 2004

Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-2 of which are triangular. Example: a(3)=6 because the convex hexagon ABCDEF is dissected by any of the diagonals AC, BD, CE, DF, EA, FB into regions containing exactly 1 triangle. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004

Number of UUU's (triple rises), where U=(1,1), in all Dyck paths of semilength n+1. Example: a(3)=6 because we have UD(UUU)DDD, (UUU)DDDUD, (UUU)DUDDD, (UUU)DDUDD, and (U[UU)U]DDDD, the triple rises being shown between parentheses. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 03 2004

Inverse binomial transform of A026389. - Ross La Haye (rlahaye(AT)new.rr.com), Mar 05 2005

Sum of the jump-lengths of all full binary trees with n internal nodes. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given full binary tree is called the jump-length. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 18 2007

a(n) = number of convex polyominoes (A005436) of perimeter 2n+4 that are directed but not parallelogram polyominoes, because the directed convex polyominoes are counted by the central binomial coefficient binom(2n,n) and the subset of parallelogram polyominoes is counted by the Catalan number C(n+1) = binom(2n+2,n+1)/(n+2), and a(n) = binom(2n,n) - C(n+1). - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007

a(n) = number of DUU's in all Dyck paths of semilength n+1. Example: a(3)=6 because we have UU(DUU)DDD, U(DUU)UDDD, U(DUU)DUDD, UDU(DUU)DD, U(DUU)DDUD, UUD(DUU)DD, the DUU's being shown between parentheses, and no other Dyck path of semilength 4 contains a DUU. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2008

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.

W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.

T. D. Noe, Table of n, a(n) for n=2..200

Milan Janjic, Two Enumerative Functions

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311

a(n) = A067310(n, 1) as this is number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 1 simple intersection. - Henry Bottomley (se16(AT)btinternet.com), Oct 07 2002

E.g.f.: exp(2*x)*BesselI(2, 2*x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 21 2003

G.f.: [1-sqrt(1-4z)]^4/[16z^2*sqrt(1-4z)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 28 2004

a(n)=sum{k=0..n, C(n, k)C(n, k+2)} - Paul Barry (pbarry(AT)wit.ie), Sep 20 2004

a:=n->sum(binomial(n, j-1)*binomial(n, j+1), j=1..n): seq(a(n), n=2..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 26 2006

Cf. A006659.

Diagonal 5 of triangle A100257.

Cf. A009766.

Sequence in context: A090777 A055715 A026031 this_sequence A007691 A065997 A006516

Adjacent sequences: A002691 A002692 A002693 this_sequence A002695 A002696 A002697

nonn

njas

More terms from Henry Bottomley (se16(AT)btinternet.com), Oct 07 2002