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A054445
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Triangle read by rows giving partial row sums of triangle A033184(n,m), n >= m >= 1 (Catalan triangle).
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3
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1, 2, 1, 5, 3, 1, 14, 9, 4, 1, 42, 28, 14, 5, 1, 132, 90, 48, 20, 6, 1, 429, 297, 165, 75, 27, 7, 1, 1430, 1001, 572, 275, 110, 35, 8, 1, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1, 16796, 11934, 7072, 3640, 1638, 637, 208, 54, 10, 1, 58786, 41990, 25194, 13260
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OFFSET
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0,2
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COMMENTS
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In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) (increasing powers of x) is (c(z)^2)/(1-x*z*c(z)) with c(z) = g.f. A000108 (Catalan numbers).
This coincides with the lower triangular Catalan convolution matrix A033184 with first row and first column deleted: a(n,m)= A033184(n+2,m+2), n >= m >= 0, a(n,m) := 0 if n<m.
The Catalan convolution matrix R(n,m) = A033184(n+1,m+1), n >= m >= 0, is the only Riordan-type matrix with R(0,0)=1 whose partial row sums (prs) matrix satisfies (prs(R))(n,m)= R(n+1,m+1), n >= m >= 0.
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LINKS
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FORMULA
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T(n, m) = Sum_{k=m..n} A033184(n+1, k+1), (partial row sums in columns m).
Column m recursion: a(n, m)= sum(a(j-1, m)*A033184(n-j+1, 1), j=m..n) + A033184(n+1, m+1) if n >= m >= 0, a(n, m) := 0 if n<m.
G.f. for column m: (c(x)^2)*(x*c(x))^m, m >= 0, with c(x) = g.f. A000108.
n-th row of the triangle = top row of M^n, where M is the following infinite square production matrix:
2, 1, 0, 0, 0, ...
1, 1, 1, 0, 0, ...
1, 1, 1, 1, 0, ...
1, 1, 1, 1, 1, ...
...
(End)
T(n, m) = (m+1) * binomial(2*n - m, n) / (n+1) if n>=m>=1. - Michael Somos, Oct 01 2018
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EXAMPLE
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Triangle starts:
1;
2, 1;
5, 3, 1;
14, 9, 4, 1;
42, 28, 14, 5, 1;
132, 90, 48, 20, 6, 1;
...
Fourth row polynomial (n=3): p(3,x)= 14 + 9*x + 4*x^2 + x^3.
Top row of M^3 = [14, 9, 4, 1, 0, 0, 0, ...].
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MATHEMATICA
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T[n_, k_] := SeriesCoefficient[((2-2*x)*y)/(2*y+x*Sqrt[1-4*y]-x), {x, 0, n}, {y, 0, k}]; Table[T[n-k+2, k], {n, 0, 10}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Apr 13 2015, after Vladimir Kruchinin *)
T[ n_, k_] := (k + 1) Binomial[2 n - k, n] / (n + 1); (* Michael Somos, Oct 01 2018 *)
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PROG
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(PARI)
tabl(nn) = {
default(seriesprecision, nn+1);
my( gf = ((2-2*x)*y)/(2*y+x*sqrt(1-4*y)-x) + O(x^nn) );
for (n=0, nn-1, my( P = polcoeff(gf, n, x) );
for (k=0, nn-1, print1(polcoeff(P, k, y), ", "); );
print(); );
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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