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%I #36 Mar 25 2024 09:59:22
%S 1,1,1,2,3,5,4,8,17,29,8,20,50,107,185,16,48,136,336,721,1257,32,112,
%T 352,968,2370,5091,8925,64,256,880,2640,7116,17304,37185,65445,128,
%U 576,2144,6928,20168,53596,129650,278635,491825,256,1280,5120,17664,54880
%N Triangle read by rows: T(n,k) = Sum_{j=0..k-1} T(n,j) + Sum_{j=1..n-k} T(n-j,k), with T(0,0)=1 and T(n,k) = 0 for k > n.
%C G.f. A(x,y) satisfies 0 = -(1-x)^2 + (1-x)(1-4x+3xy)A + 2x(1-2x-2y+3xy)A^2. G.f.: (1-x)(-(1-4x+3xy) + sqrt((1-xy)(1-9xy)))/(4x(1-2x-2y+3xy)) = 2(1-x)/(1-4x+3xy+sqrt((1-xy)(1-9xy))). - _Michael Somos_, Mar 06 2004
%C T(n,k) = number of below-diagonal lattice paths from (0,0) to (n,k) consisting of steps (k,0) (k=1,2,...) and (0,k) (k=1,2,...). Example: T(2,1)=3 because we have (1,0)(1,0)(0,1), (2,0)(0,1) and (1,0)(0,1)(1,0). - _Emeric Deutsch_, Mar 19 2004
%C T(n,k) is odd if and only if (n,k) = (0,0), k = n > 0, or k + 1 = n > 0. - _Peter Kagey_, Apr 20 2020
%D Wen-jin Woan, Diagonal lattice paths, Congressus Numerantium, 151, 2001, 173-178.
%H Peter Kagey, <a href="/A059450/b059450.txt">Table of n, a(n) for n = 0..10011</a> (141 rows flattened, first 50 rows from G. C. Greubel)
%H C. Coker, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00037-2">Enumerating a class of lattice paths</a>, Discrete Math., 271 (2003), 13-28.
%e 1;
%e 1, 1;
%e 2, 3, 5;
%e 4, 8, 17, 29;
%e 8, 20, 50, 107, 185;
%p l := 1:a[0,0] := 1:b[l] := 1:T := (n,k)->sum(a[n,j],j=0..k-1)+sum(a[n-j,k],j=1..n-k): for n from 1 to 15 do for k from 0 to n do a[n,k] := T(n,k):l := l+1:b[l] := a[n,k]: od:od:seq(b[w],w=1..l); # _Sascha Kurz_
%p # alternative
%p A059450 := proc(n,k)
%p option remember;
%p local j ;
%p if k =0 and n= 0 then
%p 1;
%p elif k > n or k < 0 then
%p 0 ;
%p else
%p add( procname(n,j),j=0..k-1) + add(procname(n-j,k),j=1..n-k) ;
%p end if;
%p end proc:
%p seq(seq(A059450(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Mar 25 2024
%t t[0, 0] = 1; t[n_, k_] /; k > n = 0; t[n_, k_] := t[n, k] = Sum[t[n, j], {j, 0, k-1}] + Sum[t[n-j, k], {j, 1, n-k}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 08 2014 *)
%o (PARI) T(n,k)=if(k<0|k>n,0,polcoeff(polcoeff(2*(1-x)/((1-4*x+3*x*y)+sqrt((1-x*y)*(1-9*x*y)+x^2*O(x^n))),n),k)) /* _Michael Somos_, Mar 06 2004 */
%o (PARI) T(n,k)=local(A,t);if(k<0|k>n,0,A=matrix(n+1,n+1);A[1,1]=1;for(m=1,n,t=0;for(j=0,m,t+=(A[m+1,j+1]=t+sum(i=1,m-j,A[m-i+1,j+1]))));A[n+1,k+1]) /* _Michael Somos_, Mar 06 2004 */
%o (PARI) T(n,k)=if(k<0|k>n,0,(n==0)+sum(j=0,k-1,T(n,j))+sum(j=1,n-k,T(n-j,k))) /* _Michael Somos_, Mar 06 2004 */
%Y Columns include A000079, A001792 (I guess), A086866, A059231. Rows sums give A086871.
%Y A059231(n) = T(n, n).
%K nonn,tabl,easy
%O 0,4
%A _N. J. A. Sloane_, Sep 16 2003
%E More terms from _Ray Chandler_, Sep 17 2003
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