A152842 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

1, 1, 3, 1, 4, 3, 1, 7, 15, 9, 1, 8, 22, 24, 9, 1, 11, 46, 90, 81, 27, 1, 12, 57, 136, 171, 108, 27, 1, 15, 93, 307, 579, 621, 351, 81, 1, 16, 108, 400, 886, 1200, 972, 432, 81, 1, 19, 156, 724, 2086, 3858, 4572, 3348, 1377, 243, 1, 20, 175, 880, 2810, 5944, 8430, 7920

Offset

0,3

Links

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

Formula

T(n,k) = T(n-1,k) + (2-(-1)^n)*T(n-1,k-1).

Sum_{k=0..n} T(n,k) = A094015(n).

T(n,n) = A108411(n+1).

T(2n,n) = A069835(n).

G.f.: (1+x+x*y)/(1-x^2-4*x^2*y-3*x^2*y^2). - Philippe Deléham , Nov 09 2013

T(n,k) = T(n-2,k) + 4*T(n-2,k-1) + 3*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 3, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

Example

The triangle T(n,k) begins:
n\k  0   1    2     3     4      5      6      7      8      9     10    11   12
0:   1
1:   1   3
2:   1   4    3
3:   1   7   15     9
4:   1   8   22    24     9
5:   1  11   46    90    81     27
6:   1  12   57   136   171    108     27
7:   1  15   93   307   579    621    351     81
8:   1  16  108   400   886   1200    972    432     81
9:   1  19  156   724  2086   3858   4572   3348   1377    243
10:  1  20  175   880  2810   5944   8430   7920   4725   1620    243
11:  1  23  235  1405  5450  14374  26262  33210  28485  15795   5103   729
12:  1  24  258  1640  6855  19824  40636  59472  61695  44280  20898  5832  729
... reformatted and extended. - Franck Maminirina Ramaharo, Feb 28 2018

Prog

(Haskell)
a152842 n k = a152842_tabl !! n !! k
a152842_row n = a152842_tabl !! n
a152842_tabl = map fst $ iterate f ([1], 3) where
   f (xs, z) = (zipWith (+) ([0] ++ map (* z) xs) (xs ++ [0]), 4 - z)
-- Reinhard Zumkeller, May 01 2014

Crossrefs

Cf. A152815, A007318, A064861.

Sequence in context: A322103 A272172 A104764 * A307280 A209704 A339106

Adjacent sequences: A152839 A152840 A152841 * A152843 A152844 A152845

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 14 2008

Status

approved