A226392 Triangle with first column identical to 1 and the other entries defined by the sum of entries above and to the left.

1, 1, 1, 1, 2, 3, 1, 4, 8, 13, 1, 8, 20, 42, 71, 1, 16, 48, 120, 256, 441, 1, 32, 112, 320, 792, 1698, 2955, 1, 64, 256, 816, 2256, 5532, 11880, 20805, 1, 128, 576, 2016, 6096, 16488, 40140, 86250, 151695, 1, 256, 1280, 4864, 15872, 46432, 123680

Offset: 0

Author

R. J. Mathar, Jun 06 2013

Keywords

• nonn
• tabl
• easy

Comments

The sequence of row sums s(n) starts at n=0 as 1, 2, 6, 26, 142, 882, 5910, 41610, 303390,... and satisfies the hypergeometric recurrence n*s(n) +2*(7-5*n)*s(n-1) +9*(n-2)*s(n-2)=0.

For n>0, s(n) = 2*T(n,n) = 2*A162326(n). - Max Alekseyev, Feb 04 2021

Examples

			T(3,2) = 8 = 3 (above) +1+4 (to the left).
1;
1,1;
1,2,3;
1,4,8,13;
1,8,20,42,71;
1,16,48,120,256,441;
1,32,112,320,792,1698,2955;
1,64,256,816,2256,5532,11880,20805;
		

Crossrefs

Cf. A162326 (diagonal), A000079 (column k=1), A001792 (column k=2).

Programs

Maple

A226392 := proc(n,k)
    option remember;
    if k = 0 then
        1;
    elif k > n or k < 0 then
        0 ;
    else
        add( procname(n,c),c=0..k-1) + add( procname(r,k),r=k..n-1) ;
    end if;
end proc:

Mathematica

t[, 0] = 1; t[n, k_] := t[n, k] = Sum[t[n, c], {c, 0, k-1}] + Sum[t[r, k], {r, k, n-1}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014 *)

Formula

Definition: T(n,0)=1. T(n,k) = sum_{0<=c0.

T(n,3) = 6*T(n-1,3) -12*T(n-2,3)+8*T(n-3,3). T(n,3) = 2^n*(n+10)*(n-1)/16.

T(n,4) = 8*T(n-1,4) -24*T(n-2,4) +32*T(n-3,4) -16*T(n-4,4); T(n,4) = 2^n*(n^2/4 +65*n/96 -47/16 +n^3/96).

For 10, T(n,n) = 2*T(n,n-1) - T(n-1,n-1). - Max Alekseyev, Feb 04 2021