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A152798 Triangle defined by T(n,k) = Sum_{j=0..k} C(k,j)*T(n-1,j+k) for n>k>0 with T(n,0)=T(n,n)=1, read by rows.

%I A152798 #7 Jun 14 2017 01:12:09
%S A152798 1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,6,6,4,1,1,1,12,15,10,5,1,1,1,27,40,
%T A152798 29,15,6,1,1,1,67,113,93,49,21,7,1,1,1,180,348,310,180,76,28,8,1,1,1,
%U A152798 528,1148,1106,685,311,111,36,9,1,1,1,1676,4045,4205,2748,1322,497,155,45
%N A152798 Triangle defined by T(n,k) = Sum_{j=0..k} C(k,j)*T(n-1,j+k) for n>k>0 with T(n,0)=T(n,n)=1, read by rows.
%e A152798 Triangle begins:
%e A152798 1;
%e A152798 1, 1;
%e A152798 1, 1, 1;
%e A152798 1, 2, 1, 1;
%e A152798 1, 3, 3, 1, 1;
%e A152798 1, 6, 6, 4, 1, 1;
%e A152798 1, 12, 15, 10, 5, 1, 1;
%e A152798 1, 27, 40, 29, 15, 6, 1, 1;
%e A152798 1, 67, 113, 93, 49, 21, 7, 1, 1;
%e A152798 1, 180, 348, 310, 180, 76, 28, 8, 1, 1;
%e A152798 1, 528, 1148, 1106, 685, 311, 111, 36, 9, 1, 1;
%e A152798 1, 1676, 4045, 4205, 2748, 1322, 497, 155, 45, 10, 1, 1;
%e A152798 1, 5721, 15203, 16912, 11683, 5858, 2323, 750, 209, 55, 11, 1, 1;
%e A152798 1, 20924, 60710, 71858, 52262, 27349, 11230, 3809, 1083, 274, 66, 12, 1, 1; ...
%e A152798 ILLUSTRATE RECURRENCE:
%e A152798 T(6,1) = T(5,1) + T(5,2) = 6 + 6 = 12;
%e A152798 T(7,2) = T(6,2) + 2*T(6,3) + T(6,4) = 6 + 2*4 + 1 = 15;
%e A152798 T(8,3) = T(7,3) + 3*T(7,4) + 3*T(7,5) + T(7,6) = 29 + 3*15 + 3*6 + 1 = 93.
%e A152798 Note that column 1 equals A122889: [1,1,2,3,6,12,27,67,180,528,...]
%e A152798 which is the antidiagonal sums of triangle A122888.
%e A152798 RELATED TRIANGLE A122888 begins:
%e A152798 1;
%e A152798 1, 1;
%e A152798 1, 2, 2, 1;
%e A152798 1, 3, 6, 9, 10, 8, 4, 1;
%e A152798 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
%e A152798 1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
%e A152798 1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
%e A152798 1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...;
%e A152798 1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...; ...
%e A152798 in which the g.f. of row n equals the n-th iteration of (x+x^2).
%o A152798 (PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, sum(j=0, k, binomial(k, j)*T(n-1, j+k))))
%Y A152798 Cf. A122888; columns: A122889, A152799; variant: A101494.
%K A152798 nonn,tabl
%O A152798 0,8
%A A152798 _Paul D. Hanna_, Dec 23 2008