A350635 Triangle read by rows: T(n,k) is the number of n-element unlabeled P-series with k connected components.

1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 10, 4, 1, 1, 31, 28, 11, 4, 1, 1, 63, 67, 31, 11, 4, 1, 1, 127, 167, 80, 32, 11, 4, 1, 1, 255, 388, 213, 83, 32, 11, 4, 1, 1, 511, 908, 534, 226, 84, 32, 11, 4, 1, 1, 1023, 2053, 1343, 580, 229, 84, 32, 11, 4, 1, 1

Offset: 1

Author

Salah Uddin Mohammad, Jan 09 2022

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1;
    1,   1;
    3,   1,  1;
    7,   4,  1,  1;
   15,  10,  4,  1,  1;
   31,  28, 11,  4,  1, 1;
   63,  67, 31, 11,  4, 1, 1;
  127, 167, 80, 32, 11, 4, 1, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums give A349276.

Column 1 is A255047(n-1).

Cf. A263864 (all posets), A349488 (disconnected).

Programs

PARI

B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x))
T(n)=[Vecrev(p/y) | p<-Vec(-1 + exp(sum(k=1, n, y^k*B(x^k)/k + O(x*x^n))))]
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 13 2022

Formula

G.f.: -1 + exp(Sum_{k>=1} y^k*B(x^k)/k) where B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x)). - Andrew Howroyd, Jan 13 2022