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).

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, 0, 1, 42, 220, 55, 0, 1, 52, 390, 286, 0, 1, 63, 637, 910, 0, 1, 75, 980, 2275, 273, 0, 1, 88, 1440, 4900, 1820, 0, 1, 102, 2040, 9520, 7140, 0, 1, 117, 2805, 17136, 21420, 1428

Offset: 0

Author

Andrew Howroyd and Janaka Rodrigo, Dec 21 2021

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  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;
  0, 1,  42,  220,   55;
  0, 1,  52,  390,  286;
  0, 1,  63,  637,  910;
  0, 1,  75,  980, 2275,  273;
  0, 1,  88, 1440, 4900, 1820;
  0, 1, 102, 2040, 9520, 7140;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1750 (rows 0..100)

Crossrefs

Columns k=2..5 are A055998, A350116, A350286, A350303.

Row sums are A114997.

Cf. A001263 (blocks of any size), A108263 (blocks of size 2 or more).

Programs

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))]}
{my(A=T(12)); for(i=1, #A, print(A[i]))}

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

Formula

G.f.: A(x,y) satisfies A(x,y) = 1 + y*(x*A(x,y))^3/(1 - x*A(x,y)).

T(n,k) = binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1) for n > 0.