A165621 Riordan array (c(x^2)*(1+xc(x^2)), xc(x^2)).
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, 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, 75, 35, 35, 9, 9, 1, 1
Offset: 0
Examples
Triangle begins 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, 6, 6, 1, 1, 14, 28, 28, 20, 20, 7, 7, 1, 1, 42, 42, 48, 48, 27, 27, 8, 8, 1, 1 The production array of this matrix begins 1, 1, 0, 0, 1, 1, 1, 0, 1, -1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, -1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1
Links
- Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Programs
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Mathematica
(* The function RiordanArray is defined in A256893. *) nmax = 10; M = PadRight[#, nmax+1]& /@ RiordanArray[(1-#)/(1-#^4)&, #/(1+#^2)&, nmax+1]; T = Inverse[M]; Table[T[[n+1, k+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
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Sage
# Algorithm of L. Seidel (1877) # Prints the first n rows of the signed version of the triangle. def Signed_A165621_triangle(n) : D = [0]*(n+4); D[1] = 1 b = False; h = 3 for i in range(2*n) : if b : for k in range(h,0,-1) : D[k] += D[k-1] h += 1 else : for k in range(1,h, 1) : D[k] -= D[k+1] if b : print([D[z] for z in (2..h-2)]) b = not b Signed_A165621_triangle(11) # Peter Luschny, May 01 2012
Formula
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,...
Comments