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A115193 Generalized Catalan triangle of Riordan type, called C(1,2). 9
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified July 24 04:25 EDT 2021. Contains 346273 sequences. (Running on oeis4.)