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A054445 Triangle read by rows giving partial row sums of triangle A033184(n,m), n >= m >= 1 (Catalan triangle). 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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<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.

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

LINKS

Table of n, a(n) for n=0..58.

FORMULA

a(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.

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

EXAMPLE

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,...]

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A132808 A135233 A125170 * A105848 A048471 A067345

Adjacent sequences:  A054442 A054443 A054444 * A054446 A054447 A054448

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang, Apr 27 2000 and May 08 2000

STATUS

approved

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Last modified January 21 17:07 EST 2019. Contains 319350 sequences. (Running on oeis4.)