A128899 Riordan array (1,(1-2x-sqrt(1-4x))/(2x)).

1, 0, 1, 0, 2, 1, 0, 5, 4, 1, 0, 14, 14, 6, 1, 0, 42, 48, 27, 8, 1, 0, 132, 165, 110, 44, 10, 1, 0, 429, 572, 429, 208, 65, 12, 1, 0, 1430, 2002, 1638, 910, 350, 90, 14, 1, 0, 4862, 7072, 6188, 3808, 1700, 544, 119, 16, 1, 0, 16796, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1

Offset: 0

Author

Philippe Deléham, Apr 21 2007

Keywords

• nonn
• tabl

Comments

Let the sequence A(n) = [0/1, 2/1, 1/2, 3/2, 2/3, 4/3, ...] defined by a(2n)=n/(n+1) and a(2n+1)=(n+2)/(n+1). T(n,k) is the triangle read by rows given by A(n) DELTA A000007 where DELTA is the operator defined in A084938.

T is the convolution triangle of the Catalan numbers (see A357368). - Peter Luschny, Oct 19 2022

Examples

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,     5,     4,     1;
  0,    14,    14,     6,     1;
  0,    42,    48,    27,     8,    1;
  0,   132,   165,   110,    44,   10,    1;
  0,   429,   572,   429,   208,   65,   12,  1;
  0,  1430,  2002,  1638,   910,  350,   90,  14,   1;
  0,  4862,  7072,  6188,  3808, 1700,  544, 119,  16,  1;
  0, 16796, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
• Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
• Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.

Crossrefs

Row sums give A088218.

Cf. A000108, A008313, A039598, A039599, A050166.

Programs

Maple

# Uses function PMatrix from A357368.
PMatrix(10, n -> binomial(2*n,n)/(n+1)); # Peter Luschny, Oct 19 2022

Mathematica

T[n_, n_] := 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n - 1, k - 1] + 2 T[n - 1, k] + T[n - 1, k + 1]; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 14 2019 *)

PARI

T(n, k) = binomial(2*n-2, n-k)-binomial(2*n-2, n-k-2); \\ Seiichi Manyama, Mar 24 2025

Sage

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(catalan_number(i)*T(k-1,n-i) for i in (1..n-k+1))
A128899 = lambda n,k: T(k,n)
for n in (0..10): print([A128899(n,k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

Formula

T(n,k) = A039598(n-1,k-1) for n >= 1, k >= 1; T(n,0)=0^n.

T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k >= 1, T(n,0)=0^n, T(n,k)=0 if k > n.

T(n,k) + T(n,k+1) = A039599(n,k). - Philippe Deléham, Sep 12 2007

Extensions

Typo in data corrected by Jean-François Alcover, Jun 14 2019