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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.

%I A152842 #23 Oct 22 2023 19:46:01
%S A152842 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,
%T A152842 108,27,1,15,93,307,579,621,351,81,1,16,108,400,886,1200,972,432,81,1,
%U A152842 19,156,724,2086,3858,4572,3348,1377,243,1,20,175,880,2810,5944,8430,7920
%N 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.
%H A152842 Reinhard Zumkeller, <a href="/A152842/b152842.txt">Rows n = 0..125 of table, flattened</a>
%F A152842 T(n,k) = T(n-1,k) + (2-(-1)^n)*T(n-1,k-1).
%F A152842 Sum_{k=0..n} T(n,k) = A094015(n).
%F A152842 T(n,n) = A108411(n+1).
%F A152842 T(2n,n) = A069835(n).
%F A152842 G.f.: (1+x+x*y)/(1-x^2-4*x^2*y-3*x^2*y^2). - _Philippe Deléham_ , Nov 09 2013
%F A152842 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
%e A152842 The triangle T(n,k) begins:
%e A152842 n\k  0   1    2     3     4      5      6      7      8      9     10    11   12
%e A152842 0:   1
%e A152842 1:   1   3
%e A152842 2:   1   4    3
%e A152842 3:   1   7   15     9
%e A152842 4:   1   8   22    24     9
%e A152842 5:   1  11   46    90    81     27
%e A152842 6:   1  12   57   136   171    108     27
%e A152842 7:   1  15   93   307   579    621    351     81
%e A152842 8:   1  16  108   400   886   1200    972    432     81
%e A152842 9:   1  19  156   724  2086   3858   4572   3348   1377    243
%e A152842 10:  1  20  175   880  2810   5944   8430   7920   4725   1620    243
%e A152842 11:  1  23  235  1405  5450  14374  26262  33210  28485  15795   5103   729
%e A152842 12:  1  24  258  1640  6855  19824  40636  59472  61695  44280  20898  5832  729
%e A152842 ... reformatted and extended. - _Franck Maminirina Ramaharo_, Feb 28 2018
%o A152842 (Haskell)
%o A152842 a152842 n k = a152842_tabl !! n !! k
%o A152842 a152842_row n = a152842_tabl !! n
%o A152842 a152842_tabl = map fst $ iterate f ([1], 3) where
%o A152842    f (xs, z) = (zipWith (+) ([0] ++ map (* z) xs) (xs ++ [0]), 4 - z)
%o A152842 -- _Reinhard Zumkeller_, May 01 2014
%Y A152842 Cf. A152815, A007318, A064861.
%K A152842 nonn,tabl
%O A152842 0,3
%A A152842 _Philippe Deléham_, Dec 14 2008