A059023 Triangle of Stirling numbers of order 4.

1, 1, 1, 1, 1, 35, 1, 126, 1, 336, 1, 792, 1, 1749, 5775, 1, 3718, 45045, 1, 7722, 231231, 1, 15808, 981981, 1, 32071, 3741738, 2627625, 1, 64702, 13307294, 35735700, 1, 130084, 45172842, 300179880, 1, 260984, 148417854, 2002016016, 1, 522937, 476330361

Offset

4,6

Comments

The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 4. This is the 4-associated Stirling number of the second kind.

This is entered as a triangular array. The entries S_4(n,k) are zero for 4k>n, so these values are omitted. Initial entry in sequence is S_4(4,1).

Rows are of lengths 1,1,1,1,2,2,2,2,3,3,3,3,...

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

Links

Alois P. Heinz, Rows n = 4..300, flattened

A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

Formula

S_r(n+1, k) = k*S_r(n, k) + binomial(n, r-1)*S_r(n-r+1, k-1); for this sequence, r=4.

G.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*t^n/n! = exp(u(e^t-sum(t^i/i!, i=0..r-1))).

T(n,k) = Sum_{j=0..min(n/3,k)} (-1)^j*n!/(6^j*j!*(n-3j)!)*S_3(n-3j,k-j), where S_3 are the 3-associated Stirling numbers of the second kind A059022. - Fabián Pereyra, Feb 21 2022

Example

There are 35 ways of partitioning a set N of cardinality 8 into 2 blocks each of cardinality at least 4, so S_4(8,2) = 35.

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      expand(x*b(n-j))*binomial(n-1, j-1), j=4..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=4..20);  # Alois P. Heinz, Feb 21 2022
# alternative
A059023 := proc(n, k)
    option remember;
    if n<4 then
        0;
    elif n < 8 and k=1 then
        1 ;
    else
        k*procname(n-1, k)+binomial(n-1, 3)*procname(n-4, k-1) ;
    end if;
end proc:  # R. J. Mathar, Apr 15 2022

Mathematica

s4[n_, k_] := k*s4[n-1, k] + Binomial[n-1, 3]*s4[n-4, k-1]; s4[n_, k_] /; 4 k > n = 0; s4[_, k_ /; k <= 0] = 0; s4[0, 0] = 1;
Flatten[Table[s4[n, k], {n, 4, 20}, {k, 1, Floor[n/4]}]][[1 ;; 42]] (* Jean-François Alcover, Jun 16 2011 *)

Crossrefs

Row sums give A057837.

Cf. A008299, A059022, A059024, A059025.

Sequence in context: A028847 A365895 A176199 * A327004 A061045 A350805

Adjacent sequences: A059020 A059021 A059022 * A059024 A059025 A059026

Keyword

nonn,tabf,nice

Author

Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

Status

approved