A054445 Triangle read by rows giving partial row sums of triangle A033184(n,m), n >= m >= 1 (Catalan triangle).

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

Offset: 0

Author

Wolfdieter Lang, Apr 27 2000 and May 08 2000

Keywords

• easy
• nonn
• tabl

Comments

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

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.

Riordan array (c(x)^2,x*c(x)) where c(x)is the g.f. of A000108. - Philippe Deléham, Nov 11 2009

Examples

			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, ...].
		

Crossrefs

Cf. A033184, A000108. Row sums: a(n+1, 1).

Programs

Mathematica

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 *)

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(); );
} \\ Michel Marcus, Apr 13 2015

Formula

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

G.f. for column m: (c(x)^2)*(x*c(x))^m, m >= 0, with c(x) = g.f. A000108.

From Gary W. Adamson, Jan 19 2012: (Start)

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)

G.f.: (((2-2*x)*y)/(2*y+x*sqrt(1-4*y)-x)-1)/(x*y). - Vladimir Kruchinin, Apr 13 2015

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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