0,3

From Benoit Cloitre, Jan 29 2002: array interpretation (first row and column are the natural numbers):

1 2 3 ..j ... if b(i,j)=b(i-1,j)+b(i-1,j-1)+b(i,j-1) then a(n+1)=b(n,n)

2 5 .........

.............

i........... b(i,j)

Number of ordered trees with 2n edges, having root of even degree, nonroot nodes of outdegree at most 2, and branches of odd length. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002

Coefficient of x^n in ((1-x)/(1-2x))^n, n>0. - Michael Somos, Sep 24 2003

Number of peaks in all Schroeder paths (i.e. consisting of steps U=(1,1), D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n,0). Example: a(2)=5 because HH,HU*D,U*DH,UHD,U*DU*D,UU*DD contain 5 peaks (indicated by *). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003

a(n) = total number of HHs in all Schroeder (n+1)-paths. Example: a(2)=5 because UH*HD, H*H*H, UDH*H, H*HUD contain 5 HHs (indicated by *) and the other 18 Schroeder 3-paths contain no HHs. - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006

G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Milan Janjic, Two Enumerative Functions

G.f.: [(1-z)/sqrt(1-6z+z^2)-1]/2 - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002

E.g.f.: exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 28 2004

a(n)=sum(k=0, n-1, binomial(n, k)*binomial(n+k+1, k+1)) - Paul Barry (pbarry(AT)wit.ie), Sep 20 2004

A064861 := proc(n, k) option remember; if n = 1 then 1; elif k = 0 then 0; else A064861(n, k-1)+(3/2-1/2*(-1)^(n+k))*A064861(n-1, k); fi; end; seq(A064861(i, i+1), i=1..40);

(PARI) a(n)=sum(k=0, n-1, binomial(n, k+1)*binomial(n+k, k))

Cf. A002003, A047781.

a(n)=T(n, n+1), array T as in A050143.

a(n)=T(n, n+1), array T as in A064861.

Half the first differences of central Delannoy numbers (A001850).

a(n)=T(n, n+1), array T as in A008288.

Sequence in context: A082308 A026718 A060928 this_sequence A102893 A094602 A048370

Adjacent sequences: A001999 A002000 A002001 this_sequence A002003 A002004 A002005

nonn

njas, Simon Plouffe (plouffe(AT)math.uqam.ca)

More terms from Clark Kimberling (ck6(AT)evansville.edu)