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

0,5

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

Links

Table of n, a(n) for n=0..51.

Formula

Definition: T(n,0)=1. T(n,k) = sum_{0<=c<k} T(n,c) + sum_{k<=r<n} T(r,k) for k>0.

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 1<k<n, T(n,k) = 2*T(n-1,k) + 2*T(n,k-1) - 3*T(n-1,k-1). For n>0, T(n,n) = 2*T(n,n-1) - T(n-1,n-1). - Max Alekseyev, Feb 04 2021

Example

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;

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 *)

Crossrefs

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

Sequence in context: A353594 A075297 A057597 * A121340 A332635 A358664

Adjacent sequences: A226389 A226390 A226391 * A226393 A226394 A226395

Keyword

nonn,tabl,easy

Author

R. J. Mathar, Jun 06 2013

Status

approved