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 A090182 Triangle T(n,k), 0<=k<=n, composed of k-Catalan numbers. 13

%I

%S 1,1,1,1,1,1,1,2,1,1,1,5,3,1,1,1,14,17,4,1,1,1,42,171,43,5,1,1,1,132,

%T 3113,1252,89,6,1,1,1,429,106419,104098,5885,161,7,1,1,1,1430,7035649,

%U 25511272,1518897,20466,265,8,1,1,1,4862,915028347,18649337311,1558435125,12833546,57799,407,9,1,1

%N Triangle T(n,k), 0<=k<=n, composed of k-Catalan numbers.

%H Alois P. Heinz, <a href="/A090182/b090182.txt">Rows n = 0..55, flattened</a>

%H Lun Lv, Zhihong Liu, <a href="https://dx.doi.org/10.1109%2FISCID.2016.1084">Some Identities Related to Restricted Lattice Paths</a>, 2016 9th International Symposium on Computational Intelligence and Design (ISCID), pp. 338-340.

%e Triangle begins :

%e 1 ;

%e 1, 1 ;

%e 1, 1, 1 ;

%e 1, 2, 1, 1 ;

%e 1, 5, 3, 1, 1 ;

%e 1, 14, 17, 4, 1, 1 ;

%e 1, 42, 171, 43, 5, 1, 1 ;

%e 1, 132, 3113, 1252, 89, 6, 1, 1 ;

%e 1, 429, 106419, 104098, 5885, 161, 7, 1, 1 ;

%e 1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1 ;

%e This sequence formatted as a square array :

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, ...

%e 1, 1, 3, 17, 171, 3113, 106419, 7035649, 915028347, 23601213721, 121358931877763, ...

%e 1, 1, 4, 43, 1252, 104098, 25511272, 18649337311, 40823535032644, 267924955577741566, ...

%e 1, 1, 5, 89, 5885, 1518897, 1558435125, 6386478643785, 104648850228298925, ...

%e 1, 1, 6, 161, 20466, 12833546, 40130703276, 627122621447281, 48995209411107768186, ...

%p T:= proc(n, k) option remember; `if`(k=n, 1, add(

%p T(j+k, k)*T(n-j-1, k)*k^j, j=0..n-k-1))

%p end:

%p seq(seq(T(n, k), k=0..n), n=0..12); # _Alois P. Heinz_, Aug 10 2017

%Y The column sequences (without leading zeros) are A000012, A000108 (Catalan), A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096 for k=0..11.

%Y T(2n,n) gives A290777.

%Y Cf. A290759.

%K easy,nonn,tabl

%O 0,8

%A _Philippe DelĂ©ham_, Jan 20 2004, Oct 16 2008

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