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

%I A052174 #25 Jul 06 2024 21:19:20
%S A052174 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 A052174 1,490,784,588,392,140,48,7,1,1764,2352,2352,1344,720,216,63,8,1,5292,
%U A052174 8820,7560,5760,2700,1215,315,80,9,1
%N A052174 Triangle of numbers arising in enumeration of walks on square lattice.
%H A052174 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 A052174 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 A052174 First few rows:
%e A052174     1;
%e A052174     1   1;
%e A052174     3   2   1;
%e A052174     6   8   3  1;
%e A052174    20  20  15  4  1;
%e A052174    50  75  45 24  5 1;
%e A052174   175 210 189 84 35 6 1;
%e A052174   ...
%t A052174 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 A052174 (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 A052174 Cf. A005558 (first column), A005559, A005560, A005561, A005562.
%K A052174 nonn,tabl,easy,nice
%O A052174 0,4
%A A052174 _N. J. A. Sloane_, Jan 26 2000