A115193 - OEIS

%I #23 Aug 28 2019 16:51:51

%S 1,1,1,3,3,1,13,13,5,1,67,67,27,7,1,381,381,157,45,9,1,2307,2307,963,

%T 291,67,11,1,14589,14589,6141,1917,477,93,13,1,95235,95235,40323,

%U 12867,3363,723,123,15,1,636925

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

%C 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).

%C 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.

%C 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).

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

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

%H Nathaniel Johnston, <a href="/A115193/b115193.txt">Table of n, a(n) for n = 0..5150</a> (up to row 100)

%H Wolfdieter Lang, <a href="/A115193/a115193.txt">First 10 rows.</a>

%F 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).

%F T(n,k) = Sum_{i=k..n} A110510(n,i) for 0 <= k <= n. - _Werner Schulte_, Mar 24 2019

%e Triangle begins:

%e    1;

%e    1,  1;

%e    3,  3,  1;

%e   13, 13,  5,  1;

%e   67, 67, 27,  7,  1;

%e   ...

%e Production matrix begins:

%e     1,   1;

%e     2,   2,   1;

%e     4,   4,   2,   1;

%e     8,   8,   4,   2,   1;

%e    16,  16,   8,   4,   2,   1;

%e    32,  32,  16,   8,   4,   2,   1;

%e    64,  64,  32,  16,   8,   4,   2,   1;

%e   128, 128,  64,  32,  16,   8,   4,   2,   1;

%e   ... _Philippe Deléham_, Sep 22 2014

%p 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

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

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

%K nonn,easy,tabl

%O 0,4

%A _Wolfdieter Lang_, Feb 23 2006