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A122896 Riordan array (1, (1 - x - sqrt(1 - 2*x - 3*x^2)) / (2*x)), a Riordan array for directed animals. Triangle read by rows.

%I A122896 #23 Oct 08 2022 06:08:01
%S A122896 1,0,1,0,1,1,0,2,2,1,0,4,5,3,1,0,9,12,9,4,1,0,21,30,25,14,5,1,0,51,76,
%T A122896 69,44,20,6,1,0,127,196,189,133,70,27,7,1,0,323,512,518,392,230,104,
%U A122896 35,8,1,0,835,1353,1422,1140,726,369,147,44,9,1
%N A122896 Riordan array (1, (1 - x - sqrt(1 - 2*x - 3*x^2)) / (2*x)), a Riordan array for directed animals. Triangle read by rows.
%C A122896 Also the convolution triangle of the Motzkin numbers A001006. - _Peter Luschny_, Oct 08 2022
%F A122896 Inverse of Riordan array (1, x / (1 + x + x^2)).
%F A122896 T(n+1, k+1) = A064189(n, k). - _Philippe Deléham_, Apr 21 2007
%F A122896 Riordan array (1, x*m(x)) where m(x) is the g.f. of Motzkin numbers (A001006). - _Philippe Deléham_, Nov 04 2009
%e A122896 Triangle begins:
%e A122896 [0] 1;
%e A122896 [1] 0,   1;
%e A122896 [2] 0,   1,   1;
%e A122896 [3] 0,   2,   2,   1;
%e A122896 [4] 0,   4,   5,   3,   1;
%e A122896 [5] 0,   9,  12,   9,   4,   1;
%e A122896 [6] 0,  21,  30,  25,  14,   5,   1;
%e A122896 [7] 0,  51,  76,  69,  44,  20,   6,  1;
%e A122896 [8] 0, 127, 196, 189, 133,  70,  27,  7, 1;
%e A122896 [9] 0, 323, 512, 518, 392, 230, 104, 35, 8, 1.
%p A122896 T := proc(n,k) option remember;
%p A122896 if k=0 then return 0^n fi; if k>n then return 0 fi;
%p A122896 T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) end:
%p A122896 for n from 0 to 9 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, Aug 17 2016
%p A122896 # Uses function PMatrix from A357368.
%p A122896 PMatrix(10, n -> simplify(hypergeom([1 -n/2, -n/2+1/2], [2], 4))); # _Peter Luschny_, Oct 08 2022
%t A122896 T[n_, n_] = 1; T[_, 0] = 0; T[n_, k_] /; 0<k<n := T[n, k] =T[n-1, k-1] + T[n-1, k] + T[n-1, k+1]; T[_, _] = 0;
%t A122896 Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* _Jean-François Alcover_, Jun 13 2019 *)
%o A122896 (Sage) # uses[riordan_array from A256893]
%o A122896 riordan_array(1, (1-x-sqrt(1-2*x-3*x^2))/(2*x), 11) # _Peter Luschny_, Aug 17 2016
%Y A122896 Row sums are A005773, number of directed animals of size n.
%Y A122896 Product of A007318 and this sequence is A122897.
%Y A122896 Cf. A001006, A007318, A064189.
%K A122896 easy,nonn,tabl
%O A122896 0,8
%A A122896 _Paul Barry_, Sep 18 2006