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A350248 Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).

%I A350248 #19 Dec 31 2021 19:59:12
%S A350248 1,0,0,0,1,0,1,0,1,0,1,3,0,1,7,0,1,12,0,1,18,12,0,1,25,45,0,1,33,110,
%T A350248 0,1,42,220,55,0,1,52,390,286,0,1,63,637,910,0,1,75,980,2275,273,0,1,
%U A350248 88,1440,4900,1820,0,1,102,2040,9520,7140,0,1,117,2805,17136,21420,1428
%N A350248 Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).
%H A350248 Andrew Howroyd, <a href="/A350248/b350248.txt">Table of n, a(n) for n = 0..1750</a> (rows 0..100)
%F A350248 G.f.: A(x,y) satisfies A(x,y) = 1 + y*(x*A(x,y))^3/(1 - x*A(x,y)).
%F A350248 T(n,k) = binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1) for n > 0.
%e A350248 Triangle begins:
%e A350248   1;
%e A350248   0;
%e A350248   0;
%e A350248   0, 1;
%e A350248   0, 1;
%e A350248   0, 1;
%e A350248   0, 1,   3;
%e A350248   0, 1,   7;
%e A350248   0, 1,  12;
%e A350248   0, 1,  18,   12;
%e A350248   0, 1,  25,   45;
%e A350248   0, 1,  33,  110;
%e A350248   0, 1,  42,  220,   55;
%e A350248   0, 1,  52,  390,  286;
%e A350248   0, 1,  63,  637,  910;
%e A350248   0, 1,  75,  980, 2275,  273;
%e A350248   0, 1,  88, 1440, 4900, 1820;
%e A350248   0, 1, 102, 2040, 9520, 7140;
%e A350248   ...
%o A350248 (PARI) T(n)={my(p=1+O(x^3)); for(i=1, n\3, p=1+y*(x*p)^3/(1-x*p)); [Vecrev(t)| t<-Vec(p + O(x*x^n))]}
%o A350248 {my(A=T(12)); for(i=1, #A, print(A[i]))}
%o A350248 (PARI) T(n,k) = if(n==0 || k>n\3, k==0, binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1)) \\ _Andrew Howroyd_, Dec 31 2021
%Y A350248 Columns k=2..5 are A055998, A350116, A350286, A350303.
%Y A350248 Row sums are A114997.
%Y A350248 Cf. A001263 (blocks of any size), A108263 (blocks of size 2 or more).
%K A350248 nonn,tabf
%O A350248 0,12
%A A350248 _Andrew Howroyd_ and _Janaka Rodrigo_,  Dec 21 2021