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A165621 Riordan array (c(x^2)*(1+xc(x^2)), xc(x^2)).

%I A165621 #25 Jun 20 2021 03:09:00
%S A165621 1,1,1,1,1,1,2,2,1,1,2,3,3,1,1,5,5,4,4,1,1,5,9,9,5,5,1,1,14,14,14,14,
%T A165621 6,6,1,1,14,28,28,20,20,7,7,1,1,42,42,48,48,27,27,8,8,1,1,42,90,90,75,
%U A165621 75,35,35,9,9,1,1
%N A165621 Riordan array (c(x^2)*(1+xc(x^2)), xc(x^2)).
%C A165621 Inverse of A165620. Row sums are A001405(n+1). Diagonal sums are A026008.
%C A165621 Factors as (1+xc(x^2),x)*(c(x^2),xc(x^2)). Transforms (-2)^n to a sequence with Hankel transform F(2n+1).
%C A165621 In general, the Hankel transform of r^n by this matrix will have a Hankel transform with g.f. (1-x)/(1+(r-1)x+x^2).
%H A165621 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry4/barry64.html">Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays</a>, JIS 12 (2009) 09.8.6.
%F A165621 Number triangle T(n,k)=sum{j=0..n, b(n-j)*sum{i=0..k, (-1)^(k-i)*C(k,i)*sum{m=0..i, C(i,m)*(C(i-m,m+k)-C(i-m,i+k+2))}}} where b(n) is the sequence beginning with 1 followed by the aerated Catalan numbers: 1,1,0,1,0,2,0,5,0,14,...
%e A165621 Triangle begins
%e A165621   1,
%e A165621   1, 1,
%e A165621   1, 1, 1,
%e A165621   2, 2, 1, 1,
%e A165621   2, 3, 3, 1, 1,
%e A165621   5, 5, 4, 4, 1, 1,
%e A165621   5, 9, 9, 5, 5, 1, 1,
%e A165621   14, 14, 14, 14, 6, 6, 1, 1,
%e A165621   14, 28, 28, 20, 20, 7, 7, 1, 1,
%e A165621   42, 42, 48, 48, 27, 27, 8, 8, 1, 1
%e A165621 The production array of this matrix begins
%e A165621   1, 1,
%e A165621   0, 0, 1,
%e A165621   1, 1, 0, 1,
%e A165621   -1, 0, 1, 0, 1,
%e A165621   1, 0, 0, 1, 0, 1,
%e A165621   -1, 0, 0, 0, 1, 0, 1,
%e A165621   1, 0, 0, 0, 0, 1, 0, 1,
%e A165621   -1, 0, 0, 0, 0, 0, 1, 0, 1,
%e A165621   1, 0, 0, 0, 0, 0, 0, 1, 0, 1
%t A165621 (* The function RiordanArray is defined in A256893. *)
%t A165621 nmax = 10;
%t A165621 M = PadRight[#, nmax+1]& /@ RiordanArray[(1-#)/(1-#^4)&, #/(1+#^2)&, nmax+1];
%t A165621 T = Inverse[M];
%t A165621 Table[T[[n+1, k+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%o A165621 (Sage) # Algorithm of L. Seidel (1877)
%o A165621 # Prints the first n rows of the signed version of the triangle.
%o A165621 def Signed_A165621_triangle(n) :
%o A165621     D = [0]*(n+4); D[1] = 1
%o A165621     b = False; h = 3
%o A165621     for i in range(2*n) :
%o A165621         if b :
%o A165621             for k in range(h,0,-1) : D[k] += D[k-1]
%o A165621             h += 1
%o A165621         else :
%o A165621             for k in range(1,h, 1) : D[k] -= D[k+1]
%o A165621         if b : print([D[z] for z in (2..h-2)])
%o A165621         b = not b
%o A165621 Signed_A165621_triangle(11) # _Peter Luschny_, May 01 2012
%K A165621 easy,nonn,tabl
%O A165621 0,7
%A A165621 _Paul Barry_, Sep 22 2009