A127898 Inverse of Riordan array (1/(1+x)^3, x/(1+x)^3).

1, 3, 1, 12, 6, 1, 55, 33, 9, 1, 273, 182, 63, 12, 1, 1428, 1020, 408, 102, 15, 1, 7752, 5814, 2565, 760, 150, 18, 1, 43263, 33649, 15939, 5313, 1265, 207, 21, 1, 246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1

Offset

0,2

Comments

The convolution triangle of A001764 (number of ternary trees). - Peter Luschny, Oct 09 2022

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8.

Formula

T(n,k) = (k+1)/(n+1)*binomial(3*n+3,n-k). - Vladimir Kruchinin, Jan 17 2013

G.f.: 1/(-y + 1/(-1 + (2*sin(1/3 *arcsin((3*sqrt(3*x))/2)))/(

sqrt(3*x))))/x. - Vladimir Kruchinin, Feb 14 2023

Example

Triangle begins:
1,
3, 1,
12, 6, 1,
55, 33, 9, 1,
273, 182, 63, 12, 1,
1428, 1020, 408, 102, 15, 1,
7752, 5814, 2565, 760, 150, 18, 1,
43263, 33649, 15939, 5313, 1265, 207, 21, 1,
246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1,
1430715, 1170585, 610740, 237510, 71253, 16443, 2842, 348, 27, 1,
8414640, 7012200, 3786588, 1553472, 503440, 129456, 26040, 3968, 432, 30, 1

Maple

# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022

Mathematica

Table[If[k == 0, Binomial[3*n, n-k]/(2*n+1), ((k+1)/n)*Binomial[3*n, n-k -1]], {n, 1, 10}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Apr 29 2018 *)

Prog

(Magma) /* As triangle: */ [[(k+1)/(n+1)*Binomial(3*n+3, n-k): k in [0..n]]: n in [0..8]];  // Bruno Berselli, Jan 17 2013
(PARI) for(n=1, 10, for(k=0, n-1, print1(if(k==0, binomial(3*n, n-k)/( 2*n +1), ((k+1)/n)*binomial(3*n, n-k-1)), ", "))) \\ G. C. Greubel, Apr 29 2018
(GAP) Flat(List([0..10], n->List([0..n], k->(k+1)/(n+1)*Binomial(3*n+3, n-k)))); # Muniru A Asiru, Apr 30 2018

Crossrefs

First column is A001764(n+1).

Row sums are A047099.

Inverse of A127895.

Sequence in context: A249253 A201638 A127894 * A078938 A135888 A329433

Adjacent sequences: A127895 A127896 A127897 * A127899 A127900 A127901

Keyword

nonn,tabl

Author

Paul Barry, Feb 04 2007

Status

approved