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A059347 Difference array of Motzkin numbers A001006 read by antidiagonals.

%I A059347 #22 Jan 08 2017 03:50:58
%S A059347 1,0,1,1,1,2,0,1,2,4,2,2,3,5,9,0,2,4,7,12,21,5,5,7,11,18,30,51,0,5,10,
%T A059347 17,28,46,76,127,14,14,19,29,46,74,120,196,323,0,14,28,47,76,122,196,
%U A059347 316,512,835,42,42,56,84,131,207,329,525,841,1353,2188,0,42,84,140,224
%N A059347 Difference array of Motzkin numbers A001006 read by antidiagonals.
%C A059347 Row sums of odd rows (e.g., 4 = 1+1+2 for 3rd row) equal the Motzkin number of next row. Row sums of even rows equal the Motzkin number of the next row - n!/((n/2)!((n/2)+1)!) (i.e., A001006(n) - A000108(n/2) where A000108 are the Catalan numbers). - _Gerald McGarvey_, Dec 05 2004
%H A059347 G. C. Greubel, <a href="/A059347/b059347.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A059347 F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.
%e A059347 Triangle begins:
%e A059347 1;
%e A059347 0,1;
%e A059347 1,1,2;
%e A059347 0,1,2,4;
%e A059347 2,2,3,5,9;
%e A059347 0,2,4,7,12,21;
%e A059347 5,5,7,11,18,30,51;
%e A059347 ...
%t A059347 max = 12; A001006 = CoefficientList[ Series[ (1-x-(1-2x-3x^2)^(1/2))/(2x^2), {x, 0, max}], x] ; row[0] = A001006; row[n_] := Differences[A001006, n]; Flatten[ Table[ row[n-k][[k]], {n, 1, max}, {k, 1, n}]] (* _Jean-François Alcover_, Nov 12 2012, from formula *)
%Y A059347 Top row is A001006, leading diagonals give A000108 (interspersed with 0's), A000108 doubled up, A059348.
%K A059347 nonn,easy,nice,tabl
%O A059347 0,6
%A A059347 _N. J. A. Sloane_, Jan 27 2001
%E A059347 More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001