A059450 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.

1, 1, 1, 2, 3, 5, 4, 8, 17, 29, 8, 20, 50, 107, 185, 16, 48, 136, 336, 721, 1257, 32, 112, 352, 968, 2370, 5091, 8925, 64, 256, 880, 2640, 7116, 17304, 37185, 65445, 128, 576, 2144, 6928, 20168, 53596, 129650, 278635, 491825, 256, 1280, 5120, 17664, 54880

Offset: 0

Author

N. J. A. Sloane, Sep 16 2003

Keywords

• nonn
• tabl
• easy

Comments

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

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

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

Examples

			1;
1,  1;
2,  3,  5;
4,  8, 17,  29;
8, 20, 50, 107, 185;
		

References

• Wen-jin Woan, Diagonal lattice paths, Congressus Numerantium, 151, 2001, 173-178.

Links

• Peter Kagey, Table of n, a(n) for n = 0..10011 (141 rows flattened, first 50 rows from G. C. Greubel)
• C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.

Crossrefs

Columns include A000079, A001792 (I guess), A086866, A059231. Rows sums give A086871.

A059231(n) = T(n, n).

Programs

Maple

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
# alternative
A059450 := proc(n,k)
    option remember;
    local j ;
    if k =0 and n= 0 then
        1;
    elif k > n or k < 0 then
        0 ;
    else
        add( procname(n,j),j=0..k-1) + add(procname(n-j,k),j=1..n-k) ;
    end if;
end proc:
seq(seq(A059450(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Mar 25 2024

Mathematica

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 *)

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 */

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 */

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 */

Extensions

More terms from Ray Chandler, Sep 17 2003