OFFSET
0,5
COMMENTS
Next term is sum of two terms above you in previous row (as in Pascal's triangle A007318) plus sum of terms directly above you on a vertical line.
T(n,n-k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), and (s,s) for s>=1. - Joerg Arndt, Jul 01 2011
Row sums gives A118649. - Emanuele Munarini, Feb 01 2017
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
G.f.: (1-x^2*y)/(1-x-x*y-2*x^2*y+x^3*y+x^3*y^2). - Emanuele Munarini, Feb 01 2017
EXAMPLE
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 5, 5, 1;
1, 7, 14, 7, 1;
1, 9, 27, 27, 9, 1;
1, 11, 44, 72, 44, 11, 1;
1, 13, 65, 149, 149, 65, 13, 1;
1, 15, 90, 266, 388, 266, 90, 15, 1;
1, 17, 119, 431, 836, 836, 431, 119, 17, 1;
MATHEMATICA
CoefficientList[#, y]& /@ CoefficientList[(1 -x^2*y)/(1 -x -x*y -2x^2*y +x^3*y + x^3*y^2) + O[x]^12, x]//Flatten (* Jean-François Alcover, Oct 30 2018, after Emanuele Munarini *)
PROG
(PARI) /* same as in A092566, but last line (output) replaced by the following */
/* show as triangle T(n-k, k): */
{ for(n=0, N-1, for(k=0, n, print1(T(n-k, k), ", "); ); print(); ); }
/* Joerg Arndt, Jul 01 2011 */
(Sage)
@CachedFunction
def T(n, k):
if (k<0): return 0
elif (k==0 or k==n): return 1
else: return + T(n-1, k-1) + T(n-1, k) + sum( T(n-2*j, k-j) for j in (1..min(k, n//2, n-k)))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 08 2021
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, May 30 2005
EXTENSIONS
More terms from Joshua Zucker, May 10 2006
Definition corrected by Emilie Hogan, Oct 15 2009
STATUS
approved