A052174 - OEIS

%I #25 Jul 06 2024 21:19:20

%S 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,

%T 1,490,784,588,392,140,48,7,1,1764,2352,2352,1344,720,216,63,8,1,5292,

%U 8820,7560,5760,2700,1215,315,80,9,1

%N Triangle of numbers arising in enumeration of walks on square lattice.

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

%F 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

%e First few rows:

%e     1;

%e     1   1;

%e     3   2   1;

%e     6   8   3  1;

%e    20  20  15  4  1;

%e    50  75  45 24  5 1;

%e   175 210 189 84 35 6 1;

%e   ...

%t 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_ *)

%o (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

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

%K nonn,tabl,easy,nice

%O 0,4

%A _N. J. A. Sloane_, Jan 26 2000