A071969 a(n) = Sum_{k=0..floor(n/3)} (binomial(n+1, k)*binomial(2*n-3*k, n-3*k)/(n+1)).

1, 1, 2, 6, 19, 63, 219, 787, 2897, 10869, 41414, 159822, 623391, 2453727, 9733866, 38877318, 156206233, 630947421, 2560537092, 10435207116, 42689715279, 175243923783, 721649457417, 2980276087005, 12340456995177, 51222441676513, 213090270498764, 888321276659112

Offset: 0

Author

N. J. A. Sloane, Jun 17 2002

Keywords

• nonn

Comments

Diagonal of A071946. - Emeric Deutsch, Dec 15 2004

Last (largest) number of each row of A071946. - David Scambler, May 15 2012

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• D. Merlini et al., Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math., 91 (1999), 197-213 (d_n page 209).

Crossrefs

Cf. A071946 is the triangle and A119254 has the row sums.

Cf. A006318, A052709, A365268, A366025.

Programs

Maple

A071969 := n->add( binomial(n+1,k)*binomial(2*n-3*k,n-3*k)/(n+1),k=0..floor(n/3));
Order:=30: g:=solve(series((H-H^2)/(1+H^3),H)=z,H): seq(coeff(g,z^n),n=1..28); # Emeric Deutsch, Dec 15 2004

Mathematica

Table[Sum[Binomial[n+1,k] Binomial[2n-3k,n-3k]/(n+1),{k,0,Floor[n/3]}],{n,0,40}] (* Harvey P. Dale, Jul 20 2022 *)

PARI

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x^3)+x^2*O(x^n)),n+1))

Formula

G.f. (offset 1) is series reversion of (x-x^2)/(1+x^3).