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A188285 Riordan matrix ( (1-2x)/(1-2x-x^2), (x-2x^2)/(1-2x-x^2) ).

%I A188285 #15 Dec 26 2023 10:07:43
%S A188285 1,0,1,1,0,1,2,2,0,1,5,4,3,0,1,12,11,6,4,0,1,29,28,18,8,5,0,1,70,72,
%T A188285 48,26,10,6,0,1,169,184,130,72,35,12,7,0,1,408,469,348,204,100,45,14,
%U A188285 8,0,1,985,1192,927,568,295,132,56,16,9,0,1,2378,3022,2456,1571,850,404,168,68,18,10,0,1,5741,7644,6477,4312,2430,1200,532,208,81,20,11,0,1
%N A188285 Riordan matrix ( (1-2x)/(1-2x-x^2), (x-2x^2)/(1-2x-x^2) ).
%C A188285 T(n,k) is the number of Dyck paths of height at most 3 with length 2n and k hills.
%C A188285 Row sum = F_(2n-1) Fibonacci number.
%C A188285 T is the convolution triangle of |A215936|. - _Peter Luschny_, Oct 19 2022
%H A188285 Vincenzo Librandi, <a href="/A188285/b188285.txt">Table of n, a(n) for n = 0..495</a>
%F A188285 T(n,k) = sum(M(i,n-k-2i)*Binomial(i+k,k)*2^{n-k-2i},i=0..floor((n-k)/2)), where M(n,k)=n(n+1)(n+2)...(n+k-1)/k!.
%F A188285 Recurrence: T(n+2,k+1) = 2 T(n+1,k+1) + T(n+1,k) + T(n,k+1) - 2 T(n,k)
%e A188285 Triangle begins:
%e A188285 1
%e A188285 0, 1
%e A188285 1, 0, 1
%e A188285 2, 2, 0, 1
%e A188285 5, 4, 3, 0, 1
%e A188285 12, 11, 6, 4, 0, 1
%e A188285 29, 28, 18, 8, 5, 0, 1
%e A188285 70, 72, 48, 26, 10, 6, 0, 1
%e A188285 169, 184, 130, 72, 35, 12, 7, 0, 1
%e A188285 408, 469, 348, 204, 100, 45, 14, 8, 0, 1
%p A188285 # Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
%p A188285 PMatrix(10, n -> (-1)^(n+1)*A215936(n)); # _Peter Luschny_, Oct 19 2022
%t A188285 Flatten[Table[Sum[Pochhammer[i,n-k-2i]/(n-k-2i)!Binomial[i+k,k]2^(n-k-2i),{i,0,(n-k)/2}],{n,0,12},{k,0,n}],1]
%o A188285 (Maxima) create_list(sum(pochhammer(i,n-k-2*i)/(n-k-2*i)!*binomial(i+k,k)*2^(n-k-2*i),i,0,(n-k)/2),n,0,12,k,0,n);
%K A188285 nonn,tabl,easy
%O A188285 0,7
%A A188285 _Emanuele Munarini_, Mar 26 2011