A185286 Triangle T(n,k) is the number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at k

1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 5, 6, 3, 3, 3, 1, 11, 11, 13, 17, 13, 7, 6, 4, 1, 24, 41, 52, 44, 43, 40, 25, 14, 10, 5, 1, 93, 120, 152, 176, 161, 126, 107, 80, 45, 25, 15, 6, 1, 272, 421, 550, 559, 561, 524, 412, 303, 227, 146, 77, 41, 21, 7, 1, 971, 1381, 1813, 2056, 2045, 1835, 1615, 1309, 938, 648, 435, 251, 126, 63, 28, 8, 1

Offset

0,5

Comments

Equivalently, the number of paths from (0,0) to (n,k) using steps of the form (1,2),(1,1),(1,-1) or (1,-2) and staying on or above the x-axis.

It appears that A047002 gives the row sums of this triangle.

Links

Robert Israel, Table of n, a(n) for n = 0..10200

Example

The table starts:
1
0,1,1
2,1,1,2,1
2,5,6,3,3,3,1

Maple

T:= proc(n, k) option remember;
  if k < 0 or k > 2*n then return 0 fi;
  procname(n-1, k-2)+procname(n-1, k-1)+procname(n-1, k+1)+procname(n-1, k+2)
end proc:
T(0, 0):= 1:
for nn from 0 to 10 do
  seq(T(nn, k), k=0..2*nn)
od; # Robert Israel, Dec 19 2017

Mathematica

T[n_, k_] := T[n, k] = If[k < 0 || k > 2n, 0, T[n-1, k-2] + T[n-1, k-1] + T[n-1, k+1] + T[n-1, k+2]];
T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Aug 19 2022, after Robert Israel *)

Prog

(PARI) flip(v)=vector(#v, i, v[#v+1-i])
ar(n)={local(p); p=1;
for(k=1, n, p*=1+x+x^3+x^4; p=(p-polcoeff(p, 0)-polcoeff(p, 1)*x)/x^2);
flip(Vec(p))}

Crossrefs

Columns k=0..2 are A187430, A055113, A296619.

Cf. A005408(row lengths), A047002(apparently row sums).

Sequence in context: A049824 A133087 A153919 * A153905 A327812 A319093

Adjacent sequences: A185283 A185284 A185285 * A185287 A185288 A185289

Keyword

nonn,walk,tabf

Author

Franklin T. Adams-Watters, Mar 10 2011

Status

approved