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

%I A115193 #27 Apr 17 2025 14:56:01
%S A115193 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 A115193 291,67,11,1,14589,14589,6141,1917,477,93,13,1,95235,95235,40323,
%U A115193 12867,3363,723,123,15,1,636925
%N A115193 Generalized Catalan triangle of Riordan type, called C(1,2).
%C A115193 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 A115193 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 A115193 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 A115193 The column sequences (without leading zeros) are A064062, A064062(n+1), A084076, A115194, A115202-A115204, for m=0..6.
%C A115193 For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.
%H A115193 Nathaniel Johnston, <a href="/A115193/b115193.txt">Table of n, a(n) for n = 0..5150</a> (up to row 100)
%H A115193 Wolfdieter Lang, <a href="/A115193/a115193.txt">First 10 rows.</a>
%F A115193 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 A115193 T(n,k) = Sum_{i=k..n} A110510(n,i) for 0 <= k <= n. - _Werner Schulte_, Mar 24 2019
%e A115193 Triangle begins:
%e A115193    1;
%e A115193    1,  1;
%e A115193    3,  3,  1;
%e A115193   13, 13,  5,  1;
%e A115193   67, 67, 27,  7,  1;
%e A115193   ...
%e A115193 Production matrix begins:
%e A115193     1,   1;
%e A115193     2,   2,   1;
%e A115193     4,   4,   2,   1;
%e A115193     8,   8,   4,   2,   1;
%e A115193    16,  16,   8,   4,   2,   1;
%e A115193    32,  32,  16,   8,   4,   2,   1;
%e A115193    64,  64,  32,  16,   8,   4,   2,   1;
%e A115193   128, 128,  64,  32,  16,   8,   4,   2,   1;
%e A115193   ... _Philippe Deléham_, Sep 22 2014
%p A115193 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
%t A115193 A110510[n_, k_] := (k/n)*Binomial[2*n - k - 1, n - k]*2^(n - k);
%t A115193 T[n_, k_] := If[n == 0, 1, Sum[A110510[n, i], {i, k, n}]];
%t A115193 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 17 2025 *)
%Y A115193 Row sums give A115197. Compare with the row reversed and scaled triangle A115195.
%Y A115193 Cf. A116866 (similar sequence C(1,3)).
%K A115193 nonn,easy,tabl
%O A115193 0,4
%A A115193 _Wolfdieter Lang_, Feb 23 2006