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A127898 Inverse of Riordan array (1/(1+x)^3, x/(1+x)^3). 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

First column is A001764(n+1). Row sums are A047099. Inverse of A127895.

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

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

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

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

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Last modified June 13 06:07 EDT 2021. Contains 344981 sequences. (Running on oeis4.)