cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A116862 Row sums of triangle A116868 (called Y(1,3)).

Original entry on oeis.org

1, 7, 55, 472, 4300, 40861, 400567, 4021720, 41147398, 427470430, 4497253054, 47817544348, 513029453530, 5547155801065, 60385754935495, 661267785740080, 7279524885630310, 80512572260496610, 894231158402339410
Offset: 0

Views

Author

Wolfdieter Lang, Mar 24 2006

Keywords

Formula

a(n)=sum(A116868(n,m),m=1..n+1), n>=0.

A116866 Generalized Catalan triangle of Riordan type, called C(1,3).

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 25, 25, 7, 1, 190, 190, 55, 10, 1, 1606, 1606, 472, 94, 13, 1, 14506, 14506, 4300, 898, 142, 16, 1, 137089, 137089, 40861, 8785, 1495, 199, 19, 1, 1338790, 1338790, 400567, 87826, 15655
Offset: 0

Views

Author

Wolfdieter Lang, Mar 24 2006

Keywords

Comments

This triangle is the second 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(a(n,m)*x^n,m=0..n) is D(x,z)=g(z)/(1 - x*z*c(3*z))= g(z)*(3*z-x*z*(1-3*z*c(3*z)))/(3*z-x*z+(x*z)^2), with g(z) and c(z) defined below.
This is the Riordan triangle named (g(x),x*c(3*x)) with g(x):=(1+3*x*c(3*x)/2)/(1+x/2) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064063 (C(3;n) Catalan generalization).
For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

Examples

			[1];[1,1];[4,4,1];[25,25,7,1];[190,190,55,10,1];...
Production matrix begins:
1, 1
3, 3, 1
9, 9, 3, 1
27, 27, 9, 3, 1
81, 81, 27, 9, 3, 1
243, 243, 81, 27, 9, 3, 1
... _Philippe Deléham_, Sep 22 2014

Links

Crossrefs

Row sums give A116867.
Compare with the row reversed and scaled triangle A116868 (called Y(1, 3)).
Cf. A115193 (similar sequence C(1,2)).

Formula

G.f. for column m>=0 is g(x)*(x*c(3*x))^m, with g(x):=(1+3*x*c(3*x)/2)/(1+x/2) and c(x) is the o.g.f. of A000108 (Catalan numbers).
Showing 1-2 of 2 results.

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