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%I #20 Nov 23 2017 23:52:40
%S 1,2,0,3,0,1,4,0,4,0,5,0,10,0,1,6,0,20,0,6,0,7,0,35,0,21,0,1,8,0,56,0,
%T 56,0,8,0,9,0,84,0,126,0,36,0,1,10,0,120,0,252,0,120,0,10,0,11,0,165,
%U 0,462,0,330,0,55,0,1,12,0,220,0,792,0,792,0,220,0,12,0
%N Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).
%C Row sums are powers of 2.
%H G. C. Greubel, <a href="/A238801/b238801.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
%F T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
%F Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.
%e Triangle begins:
%e 1;
%e 2, 0;
%e 3, 0, 1;
%e 4, 0, 4, 0;
%e 5, 0, 10, 0, 1;
%e 6, 0, 20, 0, 6, 0;
%e 7, 0, 35, 0, 21, 0, 1;
%e 8, 0, 56, 0, 56, 0, 8, 0;
%e 9, 0, 84, 0, 126, 0, 36, 0, 1;
%e 10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.
%t Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Nov 22 2017 *)
%o (PARI) T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Nov 23 2017
%Y Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.
%Y Cf. A095704, A178616.
%K nonn,tabl
%O 0,2
%A _Philippe Deléham_, Mar 05 2014
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