0,2

a(n) = number of Dyck (n+2)-paths with exactly 2 rows of peaks. A row of peaks is a maximal sequence of peaks all at the same height and 2 units apart. For example, UDUDUD ( = /\/\/\ ) contains exactly one row of peaks, as does UUUDDD, but UDUUDDUD has three and a(1)=2 counts UDUUDD, UUDDUD. - David Callan (callan(AT)stat.wisc.edu), Mar 02 2005

If X is an n-set and Y a fixed 2-subset of X then a(n-4) is equal to the number of (n-4)-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Jul 30 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797.

P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [See p. 301]

A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=4) (First published: San Francisco: Holden-Day, Inc., 1964)

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, 1972 [alternative scanned copy].

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

a(n)=binomial(n+3, n-1)+binomial(n+2, n-1).

Binomial(n,4)+2*binomial(n,3), n>=2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 2006

[seq(binomial(n, 4)+2*binomial(n, 3), n=2..43)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 2006

seq((n+4)*binomial(n, 4)/n, n=3..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 28 2007

A005582:=(-2+z)/(z-1)**5; [Conjectured by S. Plouffe in his 1992 dissertation.]

s1=s2=s3=s4=0; lst={s3}; Do[a=n+(n+2); s1+=a; s2+=s1; s3+=s2; s4+=s3; AppendTo[lst, s3/2], {n, 1, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 04 2009]

Sequence in context: A137852 A097346 A053194 this_sequence A116454 A124633 A093122

Adjacent sequences: A005579 A005580 A005581 this_sequence A005583 A005584 A005585

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000