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

Author

N. J. A. Sloane, Jan 26 2000

Keywords

• nonn
• tabl
• easy
• nice

Examples

			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;
  ...
		

Links

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

Crossrefs

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

Programs

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 *)

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

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