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A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).

%I A124725 #21 Nov 09 2019 01:11:30
%S A124725 1,1,1,2,2,1,4,4,3,1,7,8,7,4,1,11,15,15,11,5,1,16,26,30,26,16,6,1,22,
%T A124725 42,56,56,42,22,7,1,29,64,98,112,98,64,29,8,1,37,93,162,210,210,162,
%U A124725 93,37,9,1,46,130,255,372,420,372,255,130,46,10,1,56,176,385,627,792,792,627
%N A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).
%C A124725 Binomial transform of the infinite tridiagonal matrix with main diagonal, (1,1,1,...), subdiagonal, (0,0,0,...) and subsubdiagonal, (1,1,1,...). Sum of entries in row n = 2^(n+1) - n - 1 = A000325(n+1).
%C A124725 Riordan array ((1-2x+2x^2)/(1-x)^3, x/(1-x)). - _Paul Barry_, Apr 08 2011
%F A124725 T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Feb 12 2014
%e A124725 Row 3 = (4, 4, 3, 1), then row 4 = (7, 8, 7, 4, 1).
%e A124725 First few rows of the triangle are
%e A124725    1;
%e A124725    1,  1;
%e A124725    2,  2,  1;
%e A124725    4,  4,  3,  1;
%e A124725    7,  8,  7,  4,  1;
%e A124725   11, 15, 15, 11,  5,  1;
%e A124725   16, 26, 30, 26, 16,  6,  1;
%e A124725   ...
%e A124725 From _Paul Barry_, Apr 08 2011: (Start)
%e A124725 Production matrix begins
%e A124725    1, 1;
%e A124725    1, 1, 1;
%e A124725    0, 0, 1, 1;
%e A124725   -1, 0, 0, 1, 1;
%e A124725    0, 0, 0, 0, 1, 1;
%e A124725    1, 0, 0, 0, 0, 1, 1;
%e A124725    0, 0, 0, 0, 0, 0, 1, 1;
%e A124725   -1, 0, 0, 0, 0, 0, 0, 1, 1;
%e A124725    0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
%e A124725    1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
%e A124725 (End)
%p A124725 T:=(n,k)->binomial(n,k)+binomial(n,k+2): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A124725 Flatten[Table[Binomial[n,k]+Binomial[n,k+2],{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, Jun 12 2015 *)
%Y A124725 Cf. A098574, A000125, A055795, A027660, A055796, A002663.
%Y A124725 Cf. A000325.
%K A124725 nonn,tabl
%O A124725 0,4
%A A124725 _Gary W. Adamson_ and _Roger L. Bagula_, Nov 05 2006
%E A124725 Edited by _N. J. A. Sloane_, Nov 29 2006