A115193 Generalized Catalan triangle of Riordan type, called C(1,2).

1, 1, 1, 3, 3, 1, 13, 13, 5, 1, 67, 67, 27, 7, 1, 381, 381, 157, 45, 9, 1, 2307, 2307, 963, 291, 67, 11, 1, 14589, 14589, 6141, 1917, 477, 93, 13, 1, 95235, 95235, 40323, 12867, 3363, 723, 123, 15, 1, 636925

Offset

0,4

Comments

This triangle is the first of a family of generalizations of the Catalan convolution triangle A033184 (which belongs to the Bell subgroup of the Riordan group).

The o.g.f. of the row polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^n is D(x,z) = g(z)/(1 - x*z*c(2*z)) = g(z)*(2*z-x*z*(1-2*z*c(2*z)))/(2*z-x*z+(x*z)^2), with g(z) and c(z) defined below.

This is the Riordan triangle named (g(x),x*c(2*x)) with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064062 (C(2;n) Catalan generalization).

The column sequences (without leading zeros) are A064062, A064062(n+1), A084076, A115194, A115202-A115204, for m=0..6.

For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..5150 (up to row 100)

Wolfdieter Lang, First 10 rows.

Formula

G.f. for column m>=0 is g(x)*(x*c(2*x))^m, with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers).

T(n,k) = Sum_{i=k..n} A110510(n,i) for 0 <= k <= n. - Werner Schulte, Mar 24 2019

Example

Triangle begins:
   1;
   1,  1;
   3,  3,  1;
  13, 13,  5,  1;
  67, 67, 27,  7,  1;
  ...
Production matrix begins:
    1,   1;
    2,   2,   1;
    4,   4,   2,   1;
    8,   8,   4,   2,   1;
   16,  16,   8,   4,   2,   1;
   32,  32,  16,   8,   4,   2,   1;
   64,  64,  32,  16,   8,   4,   2,   1;
  128, 128,  64,  32,  16,   8,   4,   2,   1;
  ... Philippe Deléham, Sep 22 2014

Maple

lim:=7: c:=(1-sqrt(1-8*x))/(4*x): g:=(1+2*x*c)/(1+x): gf1:=g*(x*c)^m: for m from 0 to lim do t:=taylor(gf1, x, lim+1): for n from 0 to lim do a[n, m]:=coeff(t, x, n):od:od: seq(seq(a[n, m], m=0..n), n=0..lim); # Nathaniel Johnston, Apr 30 2011

Crossrefs

Row sums give A115197. Compare with the row reversed and scaled triangle A115195.

Cf. A116866 (similar sequence C(1,3)).

Sequence in context: A143603 A094021 A062746 * A227343 A216294 A039797

Adjacent sequences: A115190 A115191 A115192 * A115194 A115195 A115196

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Feb 23 2006

Status

approved