A071946 -id:A071946 - cp's OEIS frontend

A108076 Triangle in A071946 with rows reversed.

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

Offset: 0

N. J. A. Sloane, Jun 05 2005

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

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A071946.

T(n,k) = A071946(n,n-k). - David Wasserman, May 07 2008

More terms from R. J. Mathar, Aug 13 2007

More terms from David Wasserman, May 07 2008

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

Original entry on oeis.org

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).

A071943 Triangle T(n,k) (n>=0, 0 <= k <= n) 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=(1,2).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 9, 1, 4, 12, 24, 31, 1, 5, 18, 46, 89, 113, 1, 6, 25, 76, 183, 342, 431, 1, 7, 33, 115, 323, 741, 1355, 1697, 1, 8, 42, 164, 520, 1376, 3054, 5492, 6847, 1, 9, 52, 224, 786, 2326, 5900, 12768, 22669, 28161, 1, 10, 63, 296, 1134, 3684, 10370

Offset: 0

N. J. A. Sloane, Jun 15 2002

For another interpretation of this array see the Example section.

			T(3,2)=7 because we have RRRVV, RRVRV, RRVVR, RVRRV, RVRVR, RRD and RDR.
Array begins:
1,
1, 1,
1, 2, 3,
1, 3, 7, 9,
1, 4, 12, 24, 31,
1, 5, 18, 46, 89, 113,
1, 6, 25, 76, 183, 342, 431,
1, 7, 33, 115, 323, 741, 1355, 1697,
...
Equivalently, let U(n,k) (for n >= 0, 0 <= k <= n) be the number of walks from (0,0) to (n,k) using steps (1,1), (1,-1) and (0,-1). Then U(n,n-k) = T(n,k). The U(n,k) array begins:
4:  0  0  0  0  1  5 ...
3:  0  0  0  1  4 18 ...
2:  0  0  1  3 12 46 ...
1:  0  1  2  7 24 89 ...
0:  1  1  3  9 31 113 ...
-------------------------
k/n:0  1  2  3  4  5 ...
The recurrence for this version is: U(0,0)=1, U(n,k)=0 for k>n or k<0; U(n,k) = U(n,k+1) + U(n-1,k+1) + U(n,k-1). E.g. 46 = 18 + 4 + 24. Also U(n,0) = A052709(n-1).

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
James East, Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
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.
N. J. A. Sloane, Rows 0 through 100
N. J. A. Sloane, Illustration of the initial terms of the U(n,k) array

Diagonal entries yield A052709. Row sums are A071356.

Related arrays: A071944, A071945, A071946.

U:=proc(n,k) option remember;
if (n < 0) then RETURN(0);
elif (n=0) then
   if (k=0) then RETURN(1); else RETURN(0); fi;
elif (k>n or k<0) then RETURN(0);
else RETURN(U(n,k+1)+U(n-1,k+1)+U(n-1,k-1));
fi;
end;
for n from 0 to 20 do
lprint( [seq(U(n,n-i),i=0..n)] );
od:

t[0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, k] + t[n-1, k-2]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2014, after N. J. A. Sloane *)

G.f.=(1-q)/[z(2t+2t^2z-1+q)], where q=sqrt(1-4tz-4t^2z^2).

Define T(0,0)=1 and T(n,k)=0 for k<0 and k >n. Then the array is generated by the recurrence T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k-2). For example, T(5,3) = 46 = T(5,2) + T(4,3) + T(4,1) = 18 + 24 + 4. - N. J. A. Sloane, Mar 28 2013

Edited by Emeric Deutsch, Dec 21 2003

Edited by N. J. A. Sloane, Mar 28 2013

cp's OEIS Frontend

A119254 Row sums of A071946.

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A108076 Triangle in A071946 with rows reversed.

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

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A071943 Triangle T(n,k) (n>=0, 0 <= k <= n) 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=(1,2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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