A103284 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {1,1}.

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 3, 3, 4, 5, 1, 4, 5, 6, 7, 9, 1, 5, 9, 9, 11, 13, 16, 1, 6, 14, 16, 18, 20, 24, 29, 1, 7, 20, 29, 30, 34, 38, 44, 53, 1, 8, 27, 49, 53, 59, 64, 72, 82, 97, 1, 9, 35, 76, 97, 102, 112, 123, 136, 154, 179, 1, 10, 44, 111, 173, 179, 199, 214, 235, 259

Offset

0,6

Comments

Main diagonal is A103285.

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Formula

Row(n+1) = union of {1} and {T(n,k-1) + T(n,k): k=1..n}, sorted in ascending order. - Reinhard Zumkeller, Nov 19 2015

Example

Convolution of row 5 {1,4,5,6,7,9} with {1,1} = {1,5,9,11,13,16,9}; sort to obtain row 6: {1,5,9,9,11,13,16}.
Rows begin:
1,
1,1,
1,1,2,
1,2,2,3,
1,3,3,4,5,
1,4,5,6,7,9,
1,5,9,9,11,13,16,
1,6,14,16,18,20,24,29,
1,7,20,29,30,34,38,44,53,
1,8,27,49,53,59,64,72,82,97,
1,9,35,76,97,102,112,123,136,154,179,...

Prog

(PARI) {T(n, k)=local(A=vector(n+1, i, vector(i)), B); A[1][1]=1; for(k=1, n, B=vector(k+1); B[1]=1; B[k+1]=A[k][k]; for(i=2, k, B[i]=A[k][i]+A[k][i-1]); A[k+1]=vecsort(B)); return(A[n+1][k+1])}
(Haskell)
import Data.List (sort)
a103284 n k = a103284_tabl !! n !! k
a103284_row n = a103284_tabl !! n
a103284_tabl = iterate (\xs -> sort $ zipWith (+) (xs++[0]) ([0]++xs)) [1]
-- Reinhard Zumkeller, Nov 19 2015

Crossrefs

Cf. A103285.

Cf. A000079 (row sums), A257641 (central terms), A257646, A107430.

Sequence in context: A139124 A024160 A295283 * A071287 A355731 A353394

Adjacent sequences: A103281 A103282 A103283 * A103285 A103286 A103287

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 28 2005

Status

approved