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%I #25 Feb 14 2023 10:53:49
%S 1,3,1,12,6,1,55,33,9,1,273,182,63,12,1,1428,1020,408,102,15,1,7752,
%T 5814,2565,760,150,18,1,43263,33649,15939,5313,1265,207,21,1,246675,
%U 197340,98670,35880,9750,1950,273,24,1
%N Inverse of Riordan array (1/(1+x)^3, x/(1+x)^3).
%C The convolution triangle of A001764 (number of ternary trees). - _Peter Luschny_, Oct 09 2022
%H G. C. Greubel, <a href="/A127898/b127898.txt">Rows n=0..100 of triangle, flattened</a>
%H Paul Drube, <a href="https://arxiv.org/abs/2007.01892">Generalized Path Pairs and Fuss-Catalan Triangles</a>, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8.
%F T(n,k) = (k+1)/(n+1)*binomial(3*n+3,n-k). - _Vladimir Kruchinin_, Jan 17 2013
%F G.f.: 1/(-y + 1/(-1 + (2*sin(1/3 *arcsin((3*sqrt(3*x))/2)))/(
%F sqrt(3*x))))/x. - _Vladimir Kruchinin_, Feb 14 2023
%e Triangle begins:
%e 1,
%e 3, 1,
%e 12, 6, 1,
%e 55, 33, 9, 1,
%e 273, 182, 63, 12, 1,
%e 1428, 1020, 408, 102, 15, 1,
%e 7752, 5814, 2565, 760, 150, 18, 1,
%e 43263, 33649, 15939, 5313, 1265, 207, 21, 1,
%e 246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1,
%e 1430715, 1170585, 610740, 237510, 71253, 16443, 2842, 348, 27, 1,
%e 8414640, 7012200, 3786588, 1553472, 503440, 129456, 26040, 3968, 432, 30, 1
%p # Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
%p PMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # _Peter Luschny_, Oct 09 2022
%t 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 *)
%o (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
%o (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
%o (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
%Y First column is A001764(n+1).
%Y Row sums are A047099.
%Y Inverse of A127895.
%K nonn,tabl
%O 0,2
%A _Paul Barry_, Feb 04 2007
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