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A116866
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Generalized Catalan triangle of Riordan type, called C(1,3).
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2
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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.
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LINKS
| W. Lang: First 10 rows.
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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).
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EXAMPLE
| [1];[1,1];[4,4,1];[25,25,7,1];[190,190,55,10,1];...
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CROSSREFS
| Row sums give A116867.
Compare with the row reversed and scaled triangle A116868 (called Y(1, 3)).
Sequence in context: A123966 A079507 A098364 * A126280 A170986 A071207
Adjacent sequences: A116863 A116864 A116865 * A116867 A116868 A116869
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 24 2006
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