A119254 -id:A119254 - cp's OEIS frontend

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

N. J. A. Sloane, Jun 17 2002

nonn

Diagonal of A071946. - Emeric Deutsch, Dec 15 2004

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

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).

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

Cf. A006318, A052709, A365268, A366025.

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

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 *)

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

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

A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 6, 13, 19, 19, 1, 8, 23, 44, 63, 63, 1, 10, 37, 87, 156, 219, 219, 1, 12, 55, 155, 330, 568, 787, 787, 1, 14, 77, 255, 629, 1260, 2110, 2897, 2897, 1, 16, 103, 395, 1111, 2527, 4856, 7972, 10869, 10869, 1, 18, 133, 583, 1849, 4706, 10130, 18889, 30545, 41414, 41414

Offset: 0

N. J. A. Sloane, Jun 15 2002

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  2;
  1, 4,  6,  6;
  1, 6, 13, 19, 19;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Related arrays: A071943, A071944, A071945.

A108076 is the reverse, A119254 is the row sums and A071969 is the last (largest) number in each row.

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
     `if`(k<0 or nAlois P. Heinz, May 05 2023

T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
   If[k < 0 || n < k, 0, T[n-1, k] + T[n, k-1] + T[n-3, k-1]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 25 2025, after Alois P. Heinz *)

More terms from Joshua Zucker, May 10 2006

cp's OEIS Frontend

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

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A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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