A226392 - OEIS

%I #16 Feb 04 2021 16:37:24

%S 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,

%T 792,1698,2955,1,64,256,816,2256,5532,11880,20805,1,128,576,2016,6096,

%U 16488,40140,86250,151695,1,256,1280,4864,15872,46432,123680

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

%C 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.

%C For n>0, s(n) = 2*T(n,n) = 2*A162326(n). - _Max Alekseyev_, Feb 04 2021

%F 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.

%F 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.

%F 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).

%F 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

%e T(3,2) = 8 = 3 (above) +1+4 (to the left).

%e 1;

%e 1,1;

%e 1,2,3;

%e 1,4,8,13;

%e 1,8,20,42,71;

%e 1,16,48,120,256,441;

%e 1,32,112,320,792,1698,2955;

%e 1,64,256,816,2256,5532,11880,20805;

%p A226392 := proc(n,k)

%p     option remember;

%p     if k = 0 then

%p         1;

%p     elif k > n or k < 0 then

%p         0 ;

%p     else

%p         add( procname(n,c),c=0..k-1) + add( procname(r,k),r=k..n-1) ;

%p     end if;

%p end proc:

%t 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 *)

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

%K nonn,tabl,easy

%O 0,5

%A _R. J. Mathar_, Jun 06 2013