A272982 -id:A272982 - cp's OEIS frontend

A059022 Triangle of Stirling numbers of order 3.

1, 1, 1, 1, 10, 1, 35, 1, 91, 1, 210, 280, 1, 456, 2100, 1, 957, 10395, 1, 1969, 42735, 15400, 1, 4004, 158301, 200200, 1, 8086, 549549, 1611610, 1, 16263, 1827826, 10335325, 1401400, 1, 32631, 5903898, 57962905, 28028000, 1, 65382, 18682014, 297797500

Offset: 3

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 3. This is the 3-associated Stirling number of the second kind (Comtet) or the Stirling number of order 3 (Fekete).
This is entered as a triangular array. The entries S_3(n,k) are zero for 3k>n, so these values are omitted. The initial entry in the sequence is S_3(3,1).
Rows are of lengths 1,1,1,2,2,2,3,3,3,...

			There are 10 ways of partitioning a set N of cardinality 6 into 2 blocks each of cardinality at least 3, so S_3(6,2) = 10.
From _Wesley Ivan Hurt_, Feb 24 2022: (Start)
Triangle starts:
  1;
  1;
  1;
  1,   10;
  1,   35;
  1,   91;
  1,  210,   280;
  1,  456,  2100;
  1,  957, 10395;
  1, 1969, 42735, 15400;
  ...
(End)

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

Gilles Bonnet and Anna Gusakova, Concentration inequalities for Poisson U-statistics, arXiv:2404.16756 [math.PR], 2024. See p. 17.
Antal E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.
Gergő Nemes, On the Coefficients of the Asymptotic Expansion of n!, J. Int. Seq. 13 (2010), 10.6.6.

Row sums give A006505.

Cf. A008299, A059023, A059024, A059025, A100861, A272352 (column 2), A272982 (column 3), A261724 (column 4), A352611 (column 5).

b:= proc(n) option remember; `if`(n=0, 1, add(
      expand(x*b(n-j))*binomial(n-1, j-1), j=3..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=3..20);  # Alois P. Heinz, Feb 21 2022
# alternative
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:  # R. J. Mathar, Apr 15 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; Flatten[ Table[ S3[n, k], {n, 3, 20}, {k, 1, Floor[n/3]}]] (* 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=3.
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_{j=0..min(n/2,k)} (-1)^j*B(n,j)*S_2(n-2j,k-j), where B are the Bessel numbers A100861 and S_2 are the 2-associated Stirling numbers of the second kind A008299. - Fabián Pereyra, Feb 20 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

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

A352474 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into 3 intersecting polygons.

Original entry on oeis.org

268, 2055, 10285, 42515, 157911, 548912, 1826846, 5902458, 18679974, 58255005, 179762211, 550473301, 1676299353, 5083919214, 15372833564, 46383749572, 139730014800, 420448279875, 1264071072745, 3798101946855, 11406989330923, 34248214094780

Offset: 9

Janaka Rodrigo, Mar 17 2022

			The set of vertices of a convex 11-gon can be partitioned into 3 polygons in 10395 different ways:
- as 2 triangles and 1 pentagon ((1/2!)*C(11,3)*C(8,3)*C(5,5) = 4620 different ways) or
- as 1 triangle and 2 quadrilaterals ((1/2!)*C(11,3)*C(8,4)*C(4,4) = 5775 different ways).
Subtracting the A350116(11-8) = 110 nonintersecting partitions leaves a(11)=10285.

Index entries for linear recurrences with constant coefficients, signature (14,-85,294,-639,906,-839,490,-164,24).

Cf. A272982, A350116, A352477.

b(n) = if (n==8, 0, 3*b(n-1)+binomial(n-1,2)*(2^(n-4)+2-n-binomial(n-3,2)));
a(n) = b(n) - n*(n-1)*(n-7)*(n-8)/12; \\ Michel Marcus, Mar 19 2022

a(n) = b(n) - n*(n-1)*(n-7)*(n-8)/12, where b(n) = 3*b(n-1)+C(n-1,2)*(2^(n-4)+2-n-C(n-3,2)) for n > 8 and b(8) = 0. b(n) is given in A272982.
a(n) = A272982(n) - A350116(n-8).
G.f.: x^9*(268 - 1697*x + 4295*x^2 - 5592*x^3 + 4008*x^4 - 1520*x^5 + 240*x^6)/((1 - x)^5*(1 - 2*x)^3*(1 - 3*x)). - Stefano Spezia, Mar 19 2022

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A059022 Triangle of Stirling numbers of order 3.

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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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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.

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A352474 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into 3 intersecting polygons.

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