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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.
4
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
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
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Dec 14 2008
STATUS
approved