A392857 T(n, k) is the number of partitions of [n] into a set of k connected series-parallel posets.

1, 0, 1, 0, 2, 1, 0, 12, 6, 1, 0, 122, 60, 12, 1, 0, 1740, 850, 180, 20, 1, 0, 31922, 15540, 3390, 420, 30, 1, 0, 715932, 347774, 77280, 10150, 840, 42, 1, 0, 18978122, 9205308, 2071496, 282240, 25340, 1512, 56, 1, 0, 580513260, 281274858, 63878220, 8926008, 844200, 55692, 2520, 72, 1

Offset

0,5

Comments

As a counterpart to the Stirling triangles (for A058349's structure instead of sets or cycles), this triangle is of binomial type.

Links

Table of n, a(n) for n=0..54.

Andrei Z. Broder, The r-Stirling numbers, Discrete Mathematics, Vol. 49, Iss. 3 (1984), pp. 241-259.

Natalia L. Skirrow, A058349 (derivation of this sequence's formula for each column follows the same route).

Wikipedia, Binomial type.

Formula

E.g.f.: exp(y*f(x)), where f(x), the compositional inverse of x + 2*(1 - cosh(x)), is the e.g.f. of A058349.

T(n, k) = Sum_{j=0..n-k} (2*j)!/j! * C(n+j-1, k-1) * Stirling2(-j; n-k, j), where Stirling2(r; n, k) is an r-subset number, and Stirling2(-j; n-k, j) = |A199916(n-k,j)|.

T(n, 1) = A058349(n,k).

Sum_{k=1..n} T(n, k) = A048172(n).

Sum_{k=2..n} T(n, k) = A048174(n) for n >= 2.

Example

Triangle begins:
  n\k| 0      1      2     3     4   5  6 7
  ---+-------------------------------------
   0 | 1
   1 | 0      1
   2 | 0      2      1
   3 | 0     12      6     1
   4 | 0    122     60    12     1
   5 | 0   1740    850   180    20   1
   6 | 0  31922  15540  3390   420  30  1
   7 | 0 715932 347774 77280 10150 840 42 1

Prog

(Python)
from math import comb, factorial as fact
C=lambda n, k: int(k>=0) and comb(n, k) if n>=0 else (-1)**(-n-k)*C(~k, ~n) if k<0 else (-1)**k*C(k+~n, k)
rsubset=lambda r, n, k: k>=r and sum((-1)**(k-r-i)*comb(k-r, i)*(i+r)**(n-r) for i in range(k-r+1))//fact(k-r)
T=lambda n, k: sum(fact(2*j)//fact(j)*C(n+j-1, k-1)*rsubset(-j, n-k, j) for j in range(n-k+1))
(Python)
N = 7; seq = [A058349(n) for n in range(N+1)]
BellTransform(seq) # function defined in A394439.
# Peter Luschny, Mar 29 2026

Crossrefs

Cf. A048172, A058349 (formerly A048173), A048174, A199916, A394439.

Sequence in context: A324429 A119830 A268435 * A039910 A352399 A129467

Adjacent sequences: A392854 A392855 A392856 * A392858 A392859 A392860

Keyword

nonn,tabl

Author

Natalia L. Skirrow, Mar 27 2026

Status

approved