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.

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, 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, 528, 1148, 1106, 685, 311, 111, 36, 9, 1, 1, 1, 1676, 4045, 4205, 2748, 1322, 497, 155, 45

Offset

0,8

Links

Table of n, a(n) for n=0..74.

Example

Triangle begins:
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, 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, 528, 1148, 1106, 685, 311, 111, 36, 9, 1, 1;
1, 1676, 4045, 4205, 2748, 1322, 497, 155, 45, 10, 1, 1;
1, 5721, 15203, 16912, 11683, 5858, 2323, 750, 209, 55, 11, 1, 1;
1, 20924, 60710, 71858, 52262, 27349, 11230, 3809, 1083, 274, 66, 12, 1, 1; ...
ILLUSTRATE RECURRENCE:
T(6,1) = T(5,1) + T(5,2) = 6 + 6 = 12;
T(7,2) = T(6,2) + 2*T(6,3) + T(6,4) = 6 + 2*4 + 1 = 15;
T(8,3) = T(7,3) + 3*T(7,4) + 3*T(7,5) + T(7,6) = 29 + 3*15 + 3*6 + 1 = 93.
Note that column 1 equals A122889: [1,1,2,3,6,12,27,67,180,528,...]
which is the antidiagonal sums of triangle A122888.
RELATED TRIANGLE A122888 begins:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...;
1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...; ...
in which the g.f. of row n equals the n-th iteration of (x+x^2).

Prog

(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))))

Crossrefs

Cf. A122888; columns: A122889, A152799; variant: A101494.

Sequence in context: A331461 A238016 A185812 * A079115 A072906 A239062

Adjacent sequences: A152795 A152796 A152797 * A152799 A152800 A152801

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 23 2008

Status

approved