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A153342 Binomial transform of triangle A046854 (shifted).

%I A153342 #19 Mar 04 2025 10:43:25
%S A153342 1,2,0,4,1,0,8,4,1,0,16,12,5,1,0,32,32,18,6,1,0,64,80,56,25,7,1,0,128,
%T A153342 192,160,88,33,8,1,0,256,448,432,280,129,42,9,1,0,512,1024,1120,832,
%U A153342 450,180,52,10,1,0,1024,2304,2816,2352,1452,681,242,63,11,1,0
%N A153342 Binomial transform of triangle A046854 (shifted).
%C A153342 Row sums = odd indexed Fibonacci numbers.
%C A153342 Mirror image of triangle in A121462. - _Philippe Deléham_, Dec 31 2008
%C A153342 Triangle T(n,k), 0 <= k <= n, read by rows given by [2,0,0,0,0,0,0,0,0,0,0,0,...] DELTA [0,1/2,1/2,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 01 2009
%H A153342 Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Florez/florez51.html">Counting Subwords in Non-Decreasing Dyck Paths</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 8, 19.
%F A153342 Triangle read by rows, A007318 * A046854 (shifted down 1 row, inserting a "1" at (0,0).
%F A153342 G.f.: (1-y*x)/(1-2*x-y*x+y*x^2). - _Philippe Deléham_, Mar 27 2012
%F A153342 T(n,k) = 2*T(n-1,l) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 27 2012
%e A153342 First few rows of the triangle =
%e A153342     1;
%e A153342     2,    0;
%e A153342     4,    1,    0;
%e A153342     8,    4,    1,   0;
%e A153342    16,   12,    5,   1,   0;
%e A153342    32,   32,   18,   6,   1,   0;
%e A153342    64,   80,   56,  25,   7,   1,  0;
%e A153342   128,  192,  160,  88,  33,   8,  1,  0;
%e A153342   256,  448,  432, 280, 129,  42,  9,  1, 0;
%e A153342   512, 1024, 1120, 832, 450, 180, 52, 10, 1, 0;
%e A153342   ...
%Y A153342 Cf. A046854, A001519.
%K A153342 nonn,tabl
%O A153342 0,2
%A A153342 _Gary W. Adamson_, Dec 24 2008
%E A153342 Second term corrected by _Philippe Deléham_, Jan 01 2009