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

0,7

Comments

Inverse of A165620. Row sums are A001405(n+1). Diagonal sums are A026008.

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).

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).

Links

Table of n, a(n) for n=0..65.

Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.

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,...

Example

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A324029 A334046 A136605 * A284321 A004739 A156282

Adjacent sequences: A165618 A165619 A165620 * A165622 A165623 A165624

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 22 2009

Status

approved