A272352 -id:A272352 - 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

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

A201385 Irregular triangle T(n,k) (n >= 1, k >= 1) read by rows: T(n,1) = 2^n - 1; for k>1, T(n,k) = 0 for n <= 2*(k-1); otherwise T(n+1,k) = T(n,k-1) + T(n,k).

Original entry on oeis.org

1, 3, 7, 3, 15, 10, 31, 25, 10, 63, 56, 35, 127, 119, 91, 35, 255, 246, 210, 126, 511, 501, 456, 336, 126, 1023, 1012, 957, 792, 462, 2047, 2035, 1969, 1749, 1254, 462, 4095, 4082, 4004, 3718, 3003, 1716, 8191, 8177, 8086, 7722, 6721, 4719, 1716

Offset: 1

Jonathan Vos Post, Nov 30 2011

A "Pascal Staircase".

The zero entries simplify the definition, but are not part of the official triangle.

			Triangle begins:
    1
    3
    7   3
   15  10
   31  25 10
   63  56 35
  127 119 91 35
  ...

Alois P. Heinz, Rows n = 1..200, flattened
Ozer Ozturk and Piotr Pragacz, On Schur function expansions of Thom polynomials, arXiv:1111.6612 [math.AG], 2011-2012. See (59), p. 22.

Columns k = 1, 2, 3 give A000225, A000247, A272352(n+1).

Row sums give A130783.

With[{rowmax=20},DeleteCases[Transpose[PadLeft[NestWhileList[Accumulate[#[[2;;-2]]]&,2^Range[rowmax]-1,Length[#]>2&]]],0,2]] (* Paolo Xausa, Nov 07 2023 *)

Entry revised by N. J. A. Sloane, Nov 07 2023

A322291 Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).

Original entry on oeis.org

1, 2, 3, 3, 6, 7, 6, 10, 14, 15, 10, 20, 25, 30, 31, 20, 35, 50, 56, 62, 63, 35, 70, 91, 112, 119, 126, 127, 70, 126, 182, 210, 238, 246, 254, 255, 126, 252, 336, 420, 456, 492, 501, 510, 511, 252, 462, 672, 792, 912, 957, 1002, 1012, 1022, 1023, 462, 924, 1254, 1584, 1749, 1914, 1969, 2024, 2035, 2046, 2047

Offset: 1

Stefano Spezia, Aug 28 2019

T(n, k) is a sharp upper bound on the cardinality of a k-antichain in {0, 1}^n due to P. Erdős.

T(n, k) is also the total number of compositions with first part k, n+1 parts, and all differences between adjacent parts in {-1,1}. - John Tyler Rascoe, May 07 2023

			n\k|   1    2    3    4    5    6
---+-----------------------------
1  |   1
2  |   2    3
3  |   3    6    7
4  |   6   10   14   15
5  |  10   20   25   30   31
6  |  20   35   50   56   62   63
...

John Tyler Rascoe, Rows n = 1..141 of the triangle, flattened
P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51 (1945), 898-902.
C. Pelekis and V. Vlasák, On k-antichains in the unit n-cube, arXiv:1908.04727 [math.CA], 2019.

Cf. A000225 (diagonal), A189390 (row sums).

Cf. A000247, A000918, A001405, A006126, A007318, A052515, A052516, A263857, A272352, A306550.

Flat(List([1..11], n->List([1..n], k->Sum([1..k], i->Binomial(n, Int((n-k)/2)+i)))));

a:=(n, k)->sum(binomial(n, floor((1/2)*n-(1/2)*k)+i), i = 1..k): seq(seq(a(n, k), k = 1..n), n = 1..11);

T[n_,k_]:=Sum[Binomial[n,Floor[(n-k)/2]+i],{i,1,k}]; Table[T[n,k],{n,1,11},{k,1,n}]

T(n, k) = sum(i=1, k, binomial(n, floor((n-k)/2)+i));

T(n, n) = A000225(n).
T(n, n-1) = A000918(n).
T(n, n-2) = A000247(n).
T(n, n-3) = A052515(n).
T(n, n-4) = A272352(n+1).
T(n, n-5) = A052516(n).

cp's OEIS Frontend

A059022 Triangle of Stirling numbers of order 3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A201385 Irregular triangle T(n,k) (n >= 1, k >= 1) read by rows: T(n,1) = 2^n - 1; for k>1, T(n,k) = 0 for n <= 2*(k-1); otherwise T(n+1,k) = T(n,k-1) + T(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A322291 Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula