A210658 Triangle of partial sums of Catalan numbers.

1, 2, 1, 4, 3, 2, 9, 8, 7, 5, 23, 22, 21, 19, 14, 65, 64, 63, 61, 56, 42, 197, 196, 195, 193, 188, 174, 132, 626, 625, 624, 622, 617, 603, 561, 429, 2056, 2055, 2054, 2052, 2047, 2033, 1991, 1859, 1430, 6918, 6917, 6916, 6914, 6909, 6895, 6853, 6721, 6292

Offset

0,2

Comments

Diagonal elements = Catalan numbers (A000108).

First column = partial sums of Catalan numbers (A014137).

Row sums = partial sums of central binomial coefficients (A006134).

Row square-sums = A182018. - Emanuele Munarini, Apr 06 2012

Central coefficients = A210670.

Links

Vincenzo Librandi, Rows n = 0..100, flattened

Formula

Recurrence: T(n+1,k+1) = T(n,k)+C(n+1)-C(k).

G.f. (C(x)-y*C(x*y))/((1-x)*(1-y)), where C(x)=(1-sqrt(1-4x))/(2x) is the generating series for the Catalan numbers.

Example

Triangle begins:
  1,
  2,   1,
  4,   3,   2,
  9,   8,   7,   5,
  23,  22,  21,  19,  14,
  65,  64,  63,  61,  56,  42,
  197, 196, 195, 193, 188, 174, 132

Mathematica

Flatten[Table[Sum[Binomial[2i, i]/(i+1), {i, k, n}], {n, 0, 10}, {k, 0, n}]]

Prog

(Maxima) create_list(sum(binomial(2*i, i)/(i+1), i, k, n), n, 0, 10, k, 0, n);

Crossrefs

Cf. A000108, A014137, A006134, A210670.

Sequence in context: A126136 A140169 A124731 * A376572 A143122 A093067

Adjacent sequences: A210655 A210656 A210657 * A210659 A210660 A210661

Keyword

nonn,easy,tabl

Author

Emanuele Munarini, Mar 28 2012

Status

approved