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A128899 Riordan array (1,(1-2x-sqrt(1-4x))/(2x)) . 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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 .

LINKS

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

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

EXAMPLE

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;

  ...

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 *)

PROG

(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

CROSSREFS

Cf. A000108, A039598.

Sequence in context: A226783 A245972 A088391 * A155887 A113368 A066435

Adjacent sequences:  A128896 A128897 A128898 * A128900 A128901 A128902

KEYWORD

nonn,tabl,changed

AUTHOR

Philippe Deléham, Apr 21 2007

EXTENSIONS

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

STATUS

approved

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Last modified June 24 21:16 EDT 2019. Contains 324337 sequences. (Running on oeis4.)