A059022 -id:A059022 - cp's OEIS frontend

A008299 Triangle T(n,k) of associated Stirling numbers of second kind, n >= 2, 1 <= k <= floor(n/2).

1, 1, 1, 3, 1, 10, 1, 25, 15, 1, 56, 105, 1, 119, 490, 105, 1, 246, 1918, 1260, 1, 501, 6825, 9450, 945, 1, 1012, 22935, 56980, 17325, 1, 2035, 74316, 302995, 190575, 10395, 1, 4082, 235092, 1487200, 1636635, 270270, 1, 8177, 731731, 6914908, 12122110

Offset: 2

N. J. A. Sloane

T(n,k) is the number of set partitions of [n] into k blocks of size at least 2. Compare with A008277 (blocks of size at least 1) and A059022 (blocks of size at least 3). See also A200091. Reading the table by diagonals gives A134991. The row generating polynomials are the Mahler polynomials s_n(-x). See [Roman, 4.9]. - Peter Bala, Dec 04 2011
Row n gives coefficients of moments of Poisson distribution about the mean expressed as polynomials in lambda [Haight]. The coefficients of the moments about the origin are the Stirling numbers of the second kind, A008277. - N. J. A. Sloane, Jan 24 2020
Rows are of lengths 1,1,2,2,3,3,..., a pattern typical of matrices whose diagonals are rows of another lower triangular matrix--in this instance those of A134991. - Tom Copeland, May 01 2017
For a relation to decomposition of spin correlators see Table 2 of the Delfino and Vito paper. - Tom Copeland, Nov 11 2012

			There are 3 ways of partitioning a set N of cardinality 4 into 2 blocks each of cardinality at least 2, so T(4,2)=3.
Table begins:
  1;
  1;
  1,    3;
  1,   10;
  1,   25,     15;
  1,   56,    105;
  1,  119,    490,     105;
  1,  246,   1918,    1260;
  1,  501,   6825,    9450,      945;
  1, 1012,  22935,   56980,    17325;
  1, 2035,  74316,  302995,   190575,   10395;
  1, 4082, 235092, 1487200,  1636635,  270270;
  1, 8177, 731731, 6914908, 12122110, 4099095, 135135;
  ...
Reading the table by diagonals produces the triangle A134991.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
Frank Avery Haight, "Handbook of the Poisson distribution," John Wiley, 1967. See pages 6,7, but beware of errors. [Haight on page 7 gives five different ways to generate these numbers (see link)].
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
S. Roman, The Umbral Calculus, Dover Publications, New York (2005), pp. 129-130.

Vincenzo Librandi and Alois P. Heinz, Rows n = 2..200, flattened (rows n = 2..104 from Vincenzo Librandi)
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics 22(3) (2015), #P3.37.
Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, Set Partitions and Other Bell Number Enumerated Objects, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.
Gilles Bonnet and Anna Gusakova, Concentration inequalities for Poisson U-statistics, arXiv:2404.16756 [math.PR], 2024. See p. 17.
E. Rodney Canfield, J. William Helton, and Jared A. Hughes, Uniform Convergence of an Asymptotic Approximation to Associated Stirling Numbers, arXiv:2409.01489 [math.CO], 2024. See p. 12.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
Tom Copeland, Short note on Lagrange inversion
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 3.
Daniel W. Cranston and Chun-Hung Liu, Proper Conflict-free Coloring of Graphs with Large Maximum Degree, arXiv:2211.02818 [math.CO], 2022.
Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv:1104.4323 [hep-th], 2011.
Ming-Jian Ding and Jiang Zeng, Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions, arXiv:2307.00566 [math.CO], 2023.
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.
F. A. Haight, Handbook of the Poisson distribution, John Wiley, 1967 [Annotated scan of page 7 only. Note that there is an error in the table.]
Mathematics Stack Exchange, Mahler polynomials and the zeros of the incomplete gamma function, a Mathematics Stack Exchange question by Tom Copeland, Jan 06 2016.
R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See 332. From Tom Copeland, Jan 03 2016).
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
E. G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024. See p. 2.
L. M. Smiley, Completion of a Rational Function Sequence of Carlitz, arXiv:math/0006106 [math.CO], 2000.
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
Erik Vigren and Andreas Dieckmann, A New Result in Form of Finite Triple Sums for a Series from Ramanujan’s Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.
Eric Weisstein's World of Mathematics, Mahler polynomial
Wikipedia, Mahler polynomials

Rows: A000247 (k=2), A000478 (k=3), A058844 (k=4).
Row sums: A000296, diagonal: A259877.
Cf. A059022, A059023, A059024, A059025, A134991, A137375, A200091, A269939, A340556.

A008299 := proc(n,k) local i,j,t1; if k<1 or k>floor(n/2) then t1 := 0; else
t1 := add( (-1)^i*binomial(n, i)*add( (-1)^j*(k - i - j)^(n - i)/(j!*(k - i - j)!), j = 0..k - i), i = 0..k); fi; t1; end; # N. J. A. Sloane, Dec 06 2016
G:= exp(lambda*(exp(x)-1-x)):
S:= series(G,x,21):
seq(seq(coeff(coeff(S,x,n)*n!,lambda,k),k=1..floor(n/2)),n=2..20); # Robert Israel, Jan 15 2020
T := proc(n, k) option remember; if n < 0 then return 0 fi; if k = 0 then return k^n fi; k*T(n-1, k) + (n-1)*T(n-2, k-1) end:
seq(seq(T(n,k), k=1..n/2), n=2..9); # Peter Luschny, Feb 11 2021

t[n_, k_] := Sum[ (-1)^i*Binomial[n, i]*Sum[ (-1)^j*(k - i - j)^(n - i)/(j!*(k - i - j)!), {j, 0, k - i}], {i, 0, k}]; Flatten[ Table[ t[n, k], {n, 2, 14}, {k, 1, Floor[n/2]}]] (* Jean-François Alcover, Oct 13 2011, after David Wasserman *)
Table[Sum[Binomial[n, k - j] StirlingS2[n - k + j, j] (-1)^(j + k), {j, 0, k}], {n, 15}, {k, n/2}] // Flatten (* Eric W. Weisstein, Nov 13 2018 *)

{T(n, k) = if( n < 1 || 2*k > n, n==0 && k==0, sum(i=0, k, (-1)^i * binomial( n, i) * sum(j=0, k-i, (-1)^j * (k-i-j)^(n-i) / (j! * (k-i-j)!))))}; /* Michael Somos, Oct 19 2014 */

{ T(n,k) = sum(i=0,min(n,k), (-1)^i * binomial(n,i) * stirling(n-i,k-i,2) ); } /* Max Alekseyev, Feb 27 2017 */

T(n,k) = abs(A137375(n,k)).
E.g.f. with additional constant 1: exp(t*(exp(x)-1-x)) = 1 + t*x^2/2! + t*x^3/3! + (t+3*t^2)*x^4/4! + ....
Recurrence relation: T(n+1,k) = k*T(n,k) + n*T(n-1,k-1).
T(n,k) = A134991(n-k,k); A134991(n,k) = T(n+k,k).
More generally, if S_r(n,k) gives the number of set partitions of [n] into k blocks of size at least r then we have the recurrence 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=2), with associated e.g.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*(t^n/n!) = exp(u*(e^t - Sum_{i=0..r-1} t^i/i!)).
T(n,k) = Sum_{i=0..k} (-1)^i*binomial(n, i)*Sum_{j=0..k-i} (-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!). - David Wasserman, Jun 13 2007
G.f.: (R(0)-1)/(x^2*y), where R(k) = 1 - (k+1)*y*x^2/( (k+1)*y*x^2 - (1-k*x)*(1-x-k*x)/R(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 09 2013
T(n,k) = Sum_{i=0..min(n,k)} (-1)^i * binomial(n,i) * Stirling2(n-i,k-i) = Sum_{i=0..min(n,k)} (-1)^i * A007318(n,i) * A008277(n-i,k-i). - Max Alekseyev, Feb 27 2017
T(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, n-k-j) where E2(n, k) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 11 2021

Formula and cross-references from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
Edited by Peter Bala, Dec 04 2011
Edited by N. J. A. Sloane, Jan 24 2020

A006505 Number of partitions of an n-set into boxes of size >2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 11, 36, 92, 491, 2557, 11353, 60105, 362506, 2169246, 13580815, 91927435, 650078097, 4762023647, 36508923530, 292117087090, 2424048335917, 20847410586719, 185754044235873, 1711253808769653, 16272637428430152

Offset: 0

N. J. A. Sloane

J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..579 (terms 0..250 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 102
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
J. Riordan, Cached copy of paper [With permission]

Column k=2 of A293024.
Cf. A000110, A000296, A057814, A057837, A059022.
Cf. A293038.

Copy ZL := [ B,{B=Set(Set(Z, card>=3))}, labeled ]: [seq(combstruct[count](ZL, size=n), n=0..25)]; # Zerinvary Lajos, Mar 13 2007
G:={P=Set(Set(Atom,card>=3))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..25); # Zerinvary Lajos, Dec 16 2007
g:=proc(n) option remember; if n=0 then RETURN(1); fi; if n<=2 then RETURN(0); fi; if n<=5 then RETURN(x); fi; expand(x*add(binomial(n-1,i)*g(i),i=0..n-3)); end; [seq(subs(x=1,g(n)),n=0..60)]; # N. J. A. Sloane, Jul 20 2011

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp @ x - 1 - x - x^2 / 2], {x, 0, n}]] (* Michael Somos, Jul 20 2011 *)
a[0] = 1; a[n_] := n!*Sum[Sum[k!*(-1)^(m-k)*Binomial[m, k]*Sum[StirlingS2[i+k, k]* Binomial[m-k, n-m-i]*2^(-n+m+i)/(i+k)!, {i, 0, n-m}], {k, 0, m}]/m!, {m, 1, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
Table[Sum[(-1)^j * Binomial[n, j] * BellB[n-j] * 2^((j-1)/2) * HypergeometricU[(1 - j)/2, 3/2, 1/2], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Feb 09 2020 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1 - x - x^2 / 2), n))} /* Michael Somos, Jul 20 2011 */

E.g.f.: exp ( exp x - 1 - x - (1/2)*x^2 ).
a(n) = Sum_{k=1..[n/3]} A059022(n,k), n>=3. - R. J. Mathar, Nov 08 2008
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i)/ (i+k)!))/m!); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
Define polynomials g_n by g_0=1, g_1=g_2=0, g_3=g_4=g_5=x; g(n) = x*Sum_{i=0..n-3} binomial(n-1,i)*g_i; then a(n) = g_n(1). [Riordan]
a(0) = 1; a(n) = Sum_{k=0..n-3} binomial(n-1,k+2) * a(n-k-3). - Seiichi Manyama, Sep 22 2023

More terms from Christian G. Bower, Nov 09 2000

Edited by N. J. A. Sloane, Jul 20 2011

A059023 Triangle of Stirling numbers of order 4.

Original entry on oeis.org

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

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

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,...

			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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

Alois P. Heinz, Rows n = 4..300, flattened
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

Row sums give A057837.

Cf. A008299, A059022, A059024, A059025.

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

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 *)

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

A059024 Triangle of Stirling numbers of order 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 126, 1, 462, 1, 1254, 1, 3003, 1, 6721, 1, 14443, 126126, 1, 30251, 1009008, 1, 62322, 5309304, 1, 127024, 23075052, 1, 257108, 89791416, 1, 518092, 325355316, 488864376, 1, 1041029, 1122632043, 6844101264, 1, 2088043

Offset: 5

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

The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 5. This is the 5-associated Stirling number of the second kind.
This is entered as a triangular array. The entries S_5(n,k) are zero for 5k>n, so these values are omitted. Initial entry in sequence is S_5(5,1).
Rows are of lengths 1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,...

			There are 126 ways of partitioning a set N of cardinality 10 into 2 blocks each of cardinality at least 5, so S_5(10,2) = 126.
Triangle begins:
1;
1;
1;
1;
1;
1,    126;
1,    462;
1,   1254;
1,   3003;
1,   6721;
1,  14443,    126126;
1,  30251,   1009008;
1,  62322,   5309304;
1, 127024,  23075052;
1, 257108,  89791416;
1, 518092, 325355316, 488864376;
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

Alois P. Heinz, Rows n = 5..320, flattened
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

Row sums give A057814.

Cf. A008299, A059022, A059023, A059025.

T:= proc(n,k) option remember; `if`(k<1 or k>n/5, 0,
      `if`(k=1, 1, k*T(n-1, k)+binomial(n-1, 4)*T(n-5, k-1)))
    end:
seq(seq(T(n, k), k=1..n/5), n=5..25);  # Alois P. Heinz, Aug 18 2017

S5[n_ /; 5 <= n <= 9, 1] = 1; S5[n_, k_] /; 1 <= k <= Floor[n/5] := S5[n, k] = k*S5[n-1, k] + Binomial[n-1, 4]*S5[n-5, k-1]; S5[, ] = 0; Flatten[ Table[ S5[n, k], {n, 5, 25}, {k, 1, Floor[n/5]}]] (* Jean-François Alcover, Feb 21 2012 *)

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=5.
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/4,k)} (-1)^j*n!/(24^j*j!*(n-4j)!)*S_4(n-4j,k-j), where S_4 are the 4-associated Stirling numbers of the second kind A059023. - Fabián Pereyra, Feb 21 2022

A272352 a(n) is the number of ways of putting n labeled balls into 2 indistinguishable boxes such that each box contains at least 3 balls.

Original entry on oeis.org

10, 35, 91, 210, 456, 957, 1969, 4004, 8086, 16263, 32631, 65382, 130900, 261953, 524077, 1048344, 2096898, 4194027, 8388307, 16776890, 33554080, 67108485, 134217321, 268435020, 536870446, 1073741327, 2147483119, 4294966734, 8589933996, 17179868553

Offset: 6

Vincenzo Librandi, May 11 2016

			For n=6, label the balls A, B, C, D, E, and F. Then each box must contain exactly 3 balls, and the 10 ways are ABC/DEF, ABD/CEF, ABE/CDF, ABF/CDE, ACD/BEF, ACE/BDF, ACF/BDE, ADE/BCF, ADF/BCE, AEF/BCD. - _Michael B. Porter_, Jul 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 (page 16).
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A000478, A058844, A261724, A272982, column 2 of A059022.

Column k=3 of A201385 (shifted).

[(2^n-2-2*n-2*Binomial(n,2))/2: n in [6..50]];

Table[1/2 (2^n - 2 - 2 n - 2 Binomial[n, 2]), {n, 6, 40}]
LinearRecurrence[{5,-9,7,-2},{10,35,91,210},30] (* Harvey P. Dale, Mar 29 2018 *)

G.f.: x^6*(10 - 15*x + 6*x^2)/((1 - x)^3*(1 - 2*x)).
a(n) = (2^n - 2 - 2*n - 2*binomial(n, 2))/2.
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4), for n > 3.
E.g.f.: (2 - 2*exp(x) + 2*x + x^2)^2/8. - Stefano Spezia, Jul 25 2021

A059025 Triangle of Stirling numbers of order 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 462, 1, 1716, 1, 4719, 1, 11440, 1, 25883, 1, 56134, 1, 118456, 2858856, 1, 245480, 23279256, 1, 502588, 124710300, 1, 1020680, 551496660, 1, 2061709, 2181183147, 1, 4149752, 8021782197, 1, 8333153, 28051272535

Offset: 6

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

The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 6. This is the 6-associated Stirling number of the second kind.
This is entered as a triangular array. The entries S_6(n,k) are zero for 6k>n, so these values are omitted. Initial entry in sequence is S_6(6,1).
Rows are of lengths 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, ...

			There are 462 ways of partitioning a set N of cardinality 12 into 2 blocks each of cardinality at least 6, so S_6(12,2)=462.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

Michael De Vlieger, Table of n, a(n) for n = 6..13205 (rows n = 6..400, flattened).
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 5.
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

Cf. A008299, A059022, A059023, A059024.

S6[n_ /; 6 <= n <= 11, 1] = 1; S6[n_, k_] /; 1 <= k <= Floor[n/6] := S6[n, k] = k*S6[n-1, k] + Binomial[n-1, 5]*S6[n-6, k-1]; S6[, ] = 0; Flatten[ Table[ S6[n, k], {n, 6, 24}, {k, 1, Floor[n/6]}]] (* Jean-François Alcover, Feb 21 2012 *)

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=6.

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

A261724 a(n) is the number of ways of putting n labeled balls into 4 indistinguishable boxes such that each box contains at least 3 balls.

Original entry on oeis.org

15400, 200200, 1611610, 10335325, 57962905, 297797500, 1439774336, 6662393738, 29844199346, 130445781284, 559533979466, 2365296391535, 9885290914059, 40944327590760, 168389163468240, 688631376550260, 2803570746766140, 11373212443859760, 46006062639998890

Offset: 12

Vincenzo Librandi, May 17 2016

Linear recurrence signature is given by the terms of A255002 after -1. - Bruno Berselli, May 20 2016

Colin Barker, Table of n, a(n) for n = 12..1000
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 (third formula on page 16 is incorrect).
Index entries for linear recurrences with constant coefficients, signature (30, -415, 3514, -20386, 85924, -272198, 661180, -1244717, 1822478, -2068955, 1802474, -1181760, 563888, -184752, 37152, -3456).

Cf. A000478, A058844, A272352, A272982, column 4 of A059022.

[(1/12)*(-3^(n-2)*(n^2+5*n+18)+(1/64)*(2^(2*n+5)+3*2^n*(n^4+2*n^3+19*n^2+42*n+64)-16*(n^6-9*n^5+43*n^4-91*n^3+112*n^2-32*n+8))): n in [12..40]];

Table[(1/12) (-(3^(n - 2) (n^2 + 5 n + 18)) + (1/64) (2^(2 n + 5) + 3 2^n (n^4 + 2 n^3 + 19 n^2 + 42 n + 64) - 16 (n^6 - 9 n^5 + 43 n^4 - 91 n^3 + 112 n^2 - 32 n + 8))), {n, 12, 40}]

Vec(x^12*(15400 -261800*x +1996610*x^2 -9045575*x^3 +27162905*x^4 -57079715*x^5 +86268721*x^6 -94696602*x^7 +75062256*x^8 -41952000*x^9 +15705360*x^10 -3538080*x^11 +362880*x^12) / ((1 -x)^7*(1 -2*x)^5*(1 -3*x)^3*(1 -4*x)) + O(x^30)) \\ Colin Barker, May 24 2016

a(n) = (1/12)*(-3^(n - 2)*(n^2 + 5*n + 18) + (1/64)*(2^(2*n + 5) + 3*2^n*(n^4 + 2*n^3 + 19*n^2 + 42*n + 64) - 16*(n^6 - 9*n^5 + 43*n^4 - 91*n^3 + 112*n^2 - 32*n + 8))).

G.f.: x^12*(15400 -261800*x +1996610*x^2 -9045575*x^3 +27162905*x^4 -57079715*x^5 +86268721*x^6 -94696602*x^7 +75062256*x^8 -41952000*x^9 +15705360*x^10 -3538080*x^11 +362880*x^12) / ((1 -x)^7*(1 -2*x)^5*(1 -3*x)^3*(1 -4*x)). - Colin Barker, May 24 2016

Definition, data and formula corrected by Istvan Mezo and Bruno Berselli, May 20 2016

A272982 a(n) is the number of ways of putting n labeled balls into 3 indistinguishable boxes such that each box contains at least 3 balls.

Original entry on oeis.org

280, 2100, 10395, 42735, 158301, 549549, 1827826, 5903898, 18682014, 58257810, 179765973, 550478241, 1676305723, 5083927299, 15372843684, 46383762084, 139730030100, 420448298400, 1264071094975, 3798101973315, 11406989362185, 34248214131465, 102803026929030, 308533903071390

Offset: 9

Vincenzo Librandi, May 12 2016

			For n=9, label the balls A through I. The box containing ball A can contain 8*7/2 = 28 combinations of other balls. There are 6 balls for the other two boxes, so there are A272352(6) = 10 combinations for those two boxes. Thus, a(9) = 28*10 = 280. - _Michael B. Porter_, Jul 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 9..1000
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 (second formula on page 16 is incorrect).
Index entries for linear recurrences with constant coefficients, signature (14,-85,294,-639,906,-839,490,-164,24).

Cf. A000478, A058844, A261724, A272352, column 3 of A059022.

[(1/3)*(1/16)*(6*n^4-12*n^3-3*2^n*n^2+42*n^2-9*2^n*n+12*n+8*3^n-3*2^(n+3)+24): n in [9..40]];

Table[(1/3) (1/16) (6 n^4 - 12 n^3 - 3 2^n n^2 + 42 n^2 - 9 2^n n + 12 n + 8 3^n - 3 2^(n + 3) + 24), {n, 9, 40}]
CoefficientList[Series[(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5), {x, 0, 40}], x] (* Stefano Spezia, Oct 04 2018 *)

Vec(x^9*(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5) + O(x^40)) \\ Stefano Spezia, Oct 04 2018

G.f.: x^9*(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5).
a(n) = (1/3)*(1/16)*(6*n^4 - 12*n^3 - 3*2^n*n^2 + 42*n^2 - 9*2^n*n + 12*n + 8*3^n - 3*2^(n+3) + 24).
a(n) = 3*a(n-1) + C(n-1,2)*(2^(n-4) + 2 - n - C(n-3, 2)), a(n)=0, n < 9. - Vladimir Kruchinin, Oct 04 2018

Data, formulas and programs corrected for erroneous formula in Mezo's paper by Bruno Berselli, May 21 2016

A200092 The number of ways of putting n labeled items into k labeled boxes so that each box receives at least 3 objects.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 70, 1, 182, 1, 420, 1680, 1, 912, 12600, 1, 1914, 62370, 1, 3938, 256410, 369600, 1, 8008, 949806, 4804800, 1, 16172, 3297294, 38678640, 1, 32526, 10966956, 248047800, 168168000

Offset: 3

Peter Bala, Dec 04 2011

Equivalently, the number of ordered set partitions of the set [n] into k blocks of size at least three. When the boxes are unlabeled we obtain A059022.

			Table begins
  n\k |  1     2       3
  ----+-----------------
   3  |  1
   4  |  1
   5  |  1
   6  |  1    20
   7  |  1    70
   8  |  1   182
   9  |  1   420    1680
  10  |  1   912   12600
  11  |  1  1914   62370
  ...
T(6,2) = 20: The arrangements of 6 objects into 2 boxes { } and [ ] so that each box contains at least 3 items are {1,2,3}[4,5,6], {1,2,4}[3,5,6], {1,2,5}[3,4,6], {1,2,6}[3,4,5], {1,3,4}[2,5,6], {1,3,5}[2,4,6], {1,3,6}[2,4,5], {1,4,5}[2,3,6], {1,4,6}[2,3,5], {1,5,6}[2,3,4] and the 10 other possibilities where the contents of a pair of boxes are swapped.

Cf. A019538, A059022, A200091.

E.g.f. with additional constant 1: 1/(1 - t*(exp(x) - 1 - x - x^2/2!)) = 1 + t*x^3/3! + t*x^4/4! + t*x^5/5! + (t+20*t^2)*x^6/6! + ....

Recurrence relation: T(n+1,k) = k*(T(n,k) + n*(n-1)/2*T(n-2,k-1)). T(n,k) = k!*A059022(n,k).

A352611 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into 5 polygons.

Original entry on oeis.org

1401400, 28028000, 333533200, 3073270200, 24234675465, 172096749825, 1134040872965, 7069307049805, 42240545297951, 244205509154607, 1375458924105651, 7586883537988755, 41147137237012950, 220107145169421510, 1164186829638102270, 6100518487069916910

Offset: 15

Janaka Rodrigo, Mar 23 2022

			For n=17, the set of vertices of a convex 17-gon can be partitioned into 5 polygons in 333533200 different ways:
- as 4 triangles and one pentagon ((1/4!)*C(17,3)*C(14,3)*C(11,3)*C(8,3)*C(5,5) = 95295200 different ways) or
- as 3 triangles and 2 quadrilaterals ((1/3!)*(1/2!)*C(17,3)*C(14,3)*C(11,3)*C(8,4)*C(4,4) = 238238000 different ways).

Column 5 of A059022.

A059022 := proc(n,k)
    option remember;
    if n<3 then
        0;
    elif n < 6 and k=1 then
        1 ;
    else
        k*procname(n-1,k)+binomial(n-1,2)*procname(n-3,k-1) ;
    end if;
end proc:
A352611 := proc(n)
    A059022(n,5) ;
end proc:
seq(A352611(n),n=15..50) ; # R. J. Mathar, Apr 08 2022

S3[3, 1] = S3[4, 1] = S3[5, 1] = 1;
S3[n_, k_] /; 1 <= k <= Floor[n/3] := S3[n, k] = k*S3[n-1, k] + Binomial[n-1, 2]*S3[n-3, k-1];
S3[, ] = 0;
a[n_] := S3[n, 5];
Table[a[n], {n, 15, 50}] (* Jean-François Alcover, Jul 06 2022 *)

Let S(n,k) be the number of different ways to partition the set of vertices of a convex n-gon into k polygons, where each partition contains at least 3 objects (vertices).
By the k-associated Stirling numbers of second kind, it can be deduced that S(n,k) = k*S(n-1,k) + C(n-1,2)*S(n-3,k-1).
When k = 5 this gives the required formula for this particular case,
  a(n) = S(n,5) = 5*S(n-1,5) + C(n-1,2)*S(n-3,4)
  where n > 14 and S(14,5) = 0.

cp's OEIS Frontend

A008299 Triangle T(n,k) of associated Stirling numbers of second kind, n >= 2, 1 <= k <= floor(n/2).

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A006505 Number of partitions of an n-set into boxes of size >2.

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A059023 Triangle of Stirling numbers of order 4.

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A059024 Triangle of Stirling numbers of order 5.

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A272352 a(n) is the number of ways of putting n labeled balls into 2 indistinguishable boxes such that each box contains at least 3 balls.

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A059025 Triangle of Stirling numbers of order 6.

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A261724 a(n) is the number of ways of putting n labeled balls into 4 indistinguishable boxes such that each box contains at least 3 balls.