A052174 Triangle of numbers arising in enumeration of walks on square lattice.

1, 1, 1, 3, 2, 1, 6, 8, 3, 1, 20, 20, 15, 4, 1, 50, 75, 45, 24, 5, 1, 175, 210, 189, 84, 35, 6, 1, 490, 784, 588, 392, 140, 48, 7, 1, 1764, 2352, 2352, 1344, 720, 216, 63, 8, 1, 5292, 8820, 7560, 5760, 2700, 1215, 315, 80, 9, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Formula

T(n, y) equals C(n+1,k)*C(n,k) - C(n+1,k)*C(n,k-1) if n-y = 2k, else if n-y = 2k+1 equals C(n+1,k)*C(n,k+1) - C(n+1,k+1)*C(n,k-1) (using article notation). - Michel Marcus, Oct 12 2014

Example

First few rows:
    1;
    1   1;
    3   2   1;
    6   8   3  1;
   20  20  15  4  1;
   50  75  45 24  5 1;
  175 210 189 84 35 6 1;
  ...

Mathematica

c = Binomial; T[n_, m_] /; EvenQ[n-m] := (k = (n-m)/2; c[n+1, k]*c[n, k] - c[n+1, k]*c[n, k-1]); T[n_, m_] /; OddQ[n-m] := (k = (n-m-1)/2; c[n+1, k]*c[n, k+1] - c[n+1, k+1]*c[n, k-1]); Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 13 2015, after Michel Marcus *)

Prog

(PARI) tabl(nn) = {alias(C, binomial); for (n=0, nn, for (k=0, n, if (!((n-k) % 2), kk = (n-k)/2; tnk = C(n+1, kk)*C(n, kk) - C(n+1, kk)*C(n, kk-1), kk = (n-k-1)/2; tnk = C(n+1, kk)*C(n, kk+1) - C(n+1, kk+1)*C(n, kk-1)); print1(tnk, ", "); ); print(); ); } \\ Michel Marcus, Oct 12 2014

Crossrefs

Cf. A005558 (first column), A005559, A005560, A005561, A005562.

Sequence in context: A202390 A210858 A114586 * A227790 A181897 A337977

Adjacent sequences: A052171 A052172 A052173 * A052175 A052176 A052177

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane, Jan 26 2000

Status

approved