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%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
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