A071945 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using steps R=(1,0), V=(0,1) and D=(2,1).

1, 1, 1, 1, 3, 3, 1, 5, 9, 9, 1, 7, 19, 31, 31, 1, 9, 33, 73, 113, 113, 1, 11, 51, 143, 287, 431, 431, 1, 13, 73, 249, 609, 1153, 1697, 1697, 1, 15, 99, 399, 1151, 2591, 4719, 6847, 6847, 1, 17, 129, 601, 2001, 5201, 11073, 19617, 28161, 28161, 1, 19, 163, 863, 3263

Offset

0,5

Comments

Also could be titled: "Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess king from (1,1) to (n,k) in the first quadrant using only right, diagonal up-right, and diagonal up-left moves." - Peter Kagey, Apr 20 2020

Links

Peter Kagey, Table of n, a(n) for n = 0..8255 (first 128 rows, 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.

Formula

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

Example

a(3,1)=5 because we have RRRV, RRVR, RVRR, RD and DR.
Triangle begins:
1
1 1
1 3    3
1 5    9   9
1 7   19  31   31
1 9   33  73  113  113
1 11  51 143  287  431   431
1 13  73 249  609 1153  1697  1697
1 15  99 399 1151 2591  4719  6847  6847
1 17 129 601 2001 5201 11073 19617 28161 28161

Crossrefs

Diagonal entries give A052709.

Sequence in context: A339904 A208610 A193823 * A209583 A144944 A137426

Adjacent sequences: A071942 A071943 A071944 * A071946 A071947 A071948

Keyword

nonn,easy,tabl

Author

N. J. A. Sloane, Jun 15 2002

Extensions

Edited by Emeric Deutsch, Dec 21 2003

Status

approved