A008299 -id:A008299 - cp's OEIS frontend

A280202 Number of topologies on an n-set X such that for all x in X there is a y in X such that x and y are topologically indistinguishable.

1, 0, 1, 1, 10, 31, 361, 2164, 32663, 313121, 6199024, 86219497, 2225685925, 42396094690, 1414152064833, 35520966967269, 1517860883350266, 48936884016265947, 2659543345912283917, 107827798819822505332, 7409614386025588874195, 371626299919138199117981

Offset: 0

Geoffrey Critzer, Dec 28 2016

nonn

Equivalently a(n) is the number of topologies on an n-set X such that for all x in X there is a y in X such that x and y have exactly the same neighborhoods.

			a(4) = 10 because letting X = {a,b,c,d} we have the trivial topology; {{},{b,c},{a,d},X} * 3; and {{},{a,b},X} *6.

Pontus von Brömssen, Table of n, a(n) for n = 0..37
Wikipedia, Topological indistinguishability.

Column k=0 of A280192.

Cf. A001035, A008299.

E.g.f.: A(exp(x) - 1 - x) where A(x) is the e.g.f. for A001035.

a(n) = Sum_{k=0..floor(n/2)} A008299(n,k)*A001035(k).

a(19)-a(21) from Pontus von Brömssen, Apr 05 2023

A000497 S2(j,2j+2) where S2(n,k) is a 2-associated Stirling number of the second kind.

Original entry on oeis.org

1, 25, 490, 9450, 190575, 4099095, 94594500, 2343240900, 62199262125, 1764494857125, 53338158823950, 1712934942468750, 58274046742786875, 2094379201311271875, 79318164037837725000, 3157886388887074845000

Offset: 1

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
H. W. Gould, Harris Kwong, Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.
M. Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., 56 (1934), p. 87-95.

Cf. A008299, A000504.

gf := (u,t)->exp(u*(exp(t)-1-t)); S2a := j->simplify(subs(u=0,t=0,diff(gf(u,t),u$j,t$(2*j+2)))/j!); for i from 1 to 20 do S2a(i); od;
# Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000

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}]; Table[ t[2n+2, n], {n, 1, 16}  ](* Jean-François Alcover, Feb 24 2012 *)
Table[n*(n+1)*(2*n+1)*2^n*Gamma[n+3/2]/(9*Sqrt[Pi]),{n,1,20}] (* Vaclav Kotesovec, Aug 07 2013 *)

G.f.:  x*(4*x+1)*hypergeom([3, 7/2],[],2*x)+28*x^3*hypergeom([4, 9/2],[],2*x). - Mark van Hoeij, Apr 07 2013
a(n) = n*(n+1)*(2*n+1)*2^n*GAMMA(n+3/2)/(9*sqrt(Pi)). - Vaclav Kotesovec, Aug 07 2013
(2*n-1)*(n-1)*a(n) -(n+1)*(1+2*n)^2*a(n-1)=0. - R. J. Mathar, Jun 09 2018

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000

A259877 If n is even then a(n) = n!/( 2^(n/2)(n/2)! ), otherwise a(n) = n!/( 32^((n-1)/2)*((n-3)/2)! ).

Original entry on oeis.org

1, 1, 3, 10, 15, 105, 105, 1260, 945, 17325, 10395, 270270, 135135, 4729725, 2027025, 91891800, 34459425, 1964187225, 654729075, 45831035250, 13749310575, 1159525191825, 316234143225, 31623414322500, 7905853580625, 924984868933125, 213458046676875, 28887988983603750, 6190283353629375

Offset: 2

N. J. A. Sloane, Jul 09 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..501
D. L. Andrews, Letter to N. J. A. Sloane, Apr 10 1978.
D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033. See N_n.
D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033. [Annotated scanned copy]

A001147 alternating with A000457. Interlaced diagonal of A008299.

f:=proc(n) if n mod 2 = 0 then
n!/(2^(n/2)*(n/2)!) else
n!/( 3*2^((n-1)/2)*((n-3)/2)! ); fi; end;
[seq(f(n),n=2..30)];

Table[(n!/6)*2^(-n/2)*(((2^(1/2)*(1-(-1)^n))/(n/2-3/2)!)+3*(1+(-1)^n)/(n/2)!), {n, 2, 30}] (* Wesley Ivan Hurt, Jul 10 2015 *)

main(size)={v=vector(size);for(n=2, size+1, if(n%2==0, v[n-1]=n!/(2^(n/2)*(n/2)!), v[n-1]=n!/( 3*2^((n-1)/2)*((n-3)/2)!))); return(v);} /* Anders Hellström, Jul 10 2015 */

from _future_ import division
A259877_list, a = [1], 1
for n in range(2,10**2):
    a = 6*a//(n-1) if n % 2 else a*n*(n+1)//6
    A259877_list.append(a) # Chai Wah Wu, Jul 15 2015

a(n) = (n!/6)*2^(-n/2)*(((2^(1/2)*(1-(-1)^n))/(n/2-3/2)!)+3*(1+(-1)^n)/(n/2)!). - Wesley Ivan Hurt, Jul 10 2015
a(n+1) = a(n)*n*(n+1)/6 if n is even, a(n+1) = 6*a(n)/(n-1) if n is odd. - Chai Wah Wu, Jul 15 2015
a(2*n) = A001147(n), a(2*n+1) = A000457(n-1). - Yuchun Ji, Nov 02 2020

A291104 Number of maximal irredundant sets in the n X n rook graph.

Original entry on oeis.org

1, 6, 48, 632, 10130, 194292, 4730810, 145114944, 5529662802, 256094790500, 14038667879522, 890349688082736, 64160617557387338, 5183023418382933060, 464623151635449639450, 45857185726197195813632, 4951604249874284663582498, 581839639424819461006405956

Offset: 1

Eric W. Weisstein, Aug 17 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Maximal Irredundant Set
Eric Weisstein's World of Mathematics, Rook Graph

Main diagonal of A291543.

Cf. A008299, A248744, A290586.

(* Start *)
s[n_, k_] := Sum[(-1)^i Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}]
p[m_, n_, x_] := Sum[Binomial[m, k] Binomial[n, j] k! s[n - j, k - 1] j! StirlingS2[m - k, j - 1] x^(m + n - j - k), {k, 2, m - 2}, {j, 2, m - k}]
a[n_] := 2 n^n - n! + p[n, n, 1]
Array[a, 20]
(* End *)

\\ here s(n, k) is A008299, 2*n^n - n! is A248744.
s(n, k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
p(m, n, x)={sum(k=2, m-2, sum(j=2, m-k, binomial(m, k) * binomial(n, j) * k! * s(n-j, k-1) * j! * stirling(m-k, j-1, 2) * x^(m+n-j-k) ))}
a(n) = 2*n^n - n! + p(n,n,1); \\ Andrew Howroyd, Aug 25 2017

a(n) = 2*n^n - n! + Sum_{k=2..n-2} Sum_{j=2..n-k} binomial(n,k) * binomial(n,j) * k! * A008299(n-j,k-1) * j! * stirling2(n-k,j-1). - Andrew Howroyd, Aug 25 2017

Terms a(5) and beyond from Andrew Howroyd, Aug 25 2017

A291543 Array read by antidiagonals: T(m,n) = number of maximal irredundant sets in the lattice (rook) graph K_m X K_n.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 11, 11, 4, 5, 18, 48, 18, 5, 6, 27, 109, 109, 27, 6, 7, 38, 218, 632, 218, 38, 7, 8, 51, 405, 1649, 1649, 405, 51, 8, 9, 66, 724, 4192, 10130, 4192, 724, 66, 9, 10, 83, 1277, 10889, 34801, 34801, 10889, 1277, 83, 10

Offset: 1

Andrew Howroyd, Aug 25 2017

Maximal irredundant sets can be either dominating or not. The dominating maximal irredundant sets are the minimal dominating sets (A290632). The non-dominating maximal irredundant sets are those irredundant sets that have exactly one empty row and one empty column and at least one row and one column with more than one element. See note in A290586 for some additional details.

			Array begins:
=========================================================
m\n| 1  2    3     4      5       6        7         8
---|-----------------------------------------------------
1  | 1  2    3     4      5       6        7         8...
2  | 2  6   11    18     27      38       51        66...
3  | 3 11   48   109    218     405      724      1277...
4  | 4 18  109   632   1649    4192    10889     29480...
5  | 5 27  218  1649  10130   34801   116772    402673...
6  | 6 38  405  4192  34801  194292   856225   3804880...
7  | 7 51  724 10889 116772  856225  4730810  24810465...
8  | 8 66 1277 29480 402673 3804880 24810465 145114944...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Eric Weisstein's World of Mathematics, Maximal Irredundant Set
Eric Weisstein's World of Mathematics, Rook Graph

Main diagonal is A291104.

Cf. A008299, A290586, A290632, A290818.

T32[n_, k_] := n^k + k^n - Min[n, k]!*StirlingS2[Max[n, k], Min[n, k]];
T99[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}];
T[m_, n_] /; n >= m := T32[m, n] + Sum[Sum[Binomial[m, k]*Binomial[n, j]*k!*T99[n - j, k - 1]*j!*StirlingS2[m - k, j - 1], {j, 2, m - k}], {k, 2, m - 2}]; T[m_, n_] /; n < m := T[n, m];
Table[T[m - n + 1, n], {m, 1, 10}, {n, 1, m}] // Flatten(* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

\\ here s(n,k) is A008299(n,k) and b(m,n,1) is A290632(m,n).
s(n, k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
b(m, n, x) = m^n*x^n + n^m*x^m - if(n<=m, n!*x^m*stirling(m, n, 2), m!*x^n*stirling(n, m, 2));
p(m, n, x)={sum(k=2, m-2, sum(j=2, m-k, binomial(m, k) * binomial(n, j) * k! * s(n-j, k-1) * j! * stirling(m-k, j-1, 2) * x^(m+n-j-k) ))}
T(m, n) = b(m, n, 1) + p(m, n, 1);

T(m,n) = A290632(m, n) + Sum_{k=2..m-2} Sum_{j=2..m-k} binomial(m,k) * binomial(n,j) * k! * A008299(n-j,k-1) * j! * stirling2(m-k,j-1).

A352607 Triangle read by rows. T(n, k) = Bell(k)Sum_{j=0..k}(-1)^(k+j)binomial(n, n-k+j)*Stirling2(n-k+j, j) for n >= 0 and 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 6, 0, 1, 20, 0, 1, 50, 75, 0, 1, 112, 525, 0, 1, 238, 2450, 1575, 0, 1, 492, 9590, 18900, 0, 1, 1002, 34125, 141750, 49140, 0, 1, 2024, 114675, 854700, 900900, 0, 1, 4070, 371580, 4544925, 9909900, 2110185

Offset: 0

Peter Luschny and Mélika Tebni, Mar 23 2022

			Triangle starts:
[0] 1;
[1] 0;
[2] 0, 1;
[3] 0, 1;
[4] 0, 1,   6;
[5] 0, 1,  20;
[6] 0, 1,  50,   75;
[7] 0, 1, 112,  525;
[8] 0, 1, 238, 2450,  1575;
[9] 0, 1, 492, 9590, 18900;

Cf. A028248 (row sums), A052515 (column 2), A081066, A008299, A000110, A137375.

A352607 := (n, k) -> combinat:-bell(k)*add((-1)^(k+j)*binomial(n, n-k+j)* Stirling2(n-k+j, j), j = 0..k):
seq(seq(A352607(n, k), k = 0..n/2), n = 0..12);
# Second program:
egf := k -> combinat[bell](k)*(exp(x) - 1 - x)^k/k!:
A352607 := (n, k) -> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A352607(n, k), k = 0..n/2)), n=0..12);
# Recurrence:
A352607 := proc(n, k) option remember;
if k > n/2 then 0 elif k = 0 then k^n else k*A352607(n-1, k) +
combinat[bell](k)/combinat[bell](k-1)*(n-1)*A352607(n-2, k-1) fi end:
seq(print(seq(A352607(n, k), k=0..n/2)), n=0..12); # Mélika Tebni, Mar 24 2022

T[n_, k_] := BellB[k]*Sum[(-1)^(k+j)*Binomial[n, n-k+j]*StirlingS2[n-k+j, j], {j, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Oct 21 2023 *)

T(n, k) = (-1)^k*A000110(k)*A137375(n, k) = A000110(k)*A008299(n, k).
T(2*n, n) = A081066(n).
E.g.f. column k: Bell(k)*(exp(x) - 1 - x)^k / k!, k >= 0.
T(n, k) = Bell(k)*Sum_{j=0..k} Sum_{i=0..j} ((-1)^j*(k-j)^(n-i)*binomial(n, i)) / ((k - j)!*(j - i)!).

A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 10, 0, 0, 0, 0, 1, 25, 15, 0, 0, 0, 0, 1, 56, 105, 0, 0, 0, 0, 0, 1, 119, 490, 105, 0, 0, 0, 0, 0, 1, 246, 1918, 1260, 0, 0, 0, 0, 0, 0, 1, 501, 6825, 9450, 945, 0, 0, 0, 0, 0, 0, 1, 1012, 22935, 56980, 17325, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Nov 25 2022

{{n, k}} are the number of k-quotient sets of an n-set having at least two elements in each equivalence class. This is the definition and notation (doubling the stacked delimiters of the Stirling set numbers) as given by Fekete (see link).
The formal definition expresses the second order Stirling set numbers as a binomial sum over second order Eulerian numbers (see the first formula below). The terminology 'associated Stirling numbers of second kind' used elsewhere should be dropped in favor of the more systematic one used here.
Also the Bell transform of sign(n) for n >= 0. For the definition of the Bell transform see A264428.

			Triangle T(n, k) starts:
[0] 1;
[1] 0, 0;
[2] 0, 1,   0;
[3] 0, 1,   0,    0;
[4] 0, 1,   3,    0,    0;
[5] 0, 1,  10,    0,    0,  0;
[6] 0, 1,  25,   15,    0,  0,  0;
[7] 0, 1,  56,  105,    0,  0,  0,  0;
[8] 0, 1, 119,  490,  105,  0,  0,  0,  0;
[9] 0, 1, 246, 1918, 1260,  0,  0,  0,  0,  0;

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994, thirty-fourth printing 2022.

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

A008299 is an irregular subtriangle with more information.
A358622 (second order Stirling cycle numbers).
Cf. A000296 (row sums), alternating row sums (apart from sign): A000587, A293037, and A014182.
Cf. A048993, A201637, A264428, A269939, A340264.

T := (n, k) -> add(binomial(n, k - j)*Stirling2(n - k + j, j)*(-1)^(k - j),
j = 0..k): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Using the e.g.f.:
egf := exp(z*(exp(t) - t - 1)): ser := series(egf, t, 12):
seq(print(seq(n!*coeff(coeff(ser, t, n), z, k), k = 0..n)), n = 0..9);
# Using second order Eulerian numbers:
A358623 := proc(n, k) if n = 0 then return 1 fi;
add(binomial(j, n - 2*k)*combinat:-eulerian2(n - k, n - k - j - 1), j = 0..n-k-1)
end: seq(seq(A358623(n, k), k = 0..n), n = 0..11);

# recursion over rows
from functools import cache
@cache
def StirlingSetOrd2(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 0]
    rov: list[int] = StirlingSetOrd2(n - 2)
    row: list[int] = StirlingSetOrd2(n - 1) + [0]
    for k in range(1, n // 2 + 1):
        row[k] = (n - 1) * rov[k - 1] + k * row[k]
    return row
for n in range(9): print(StirlingSetOrd2(n))
# Alternative, using function BellMatrix from A264428.
def f(k: int) -> int:
    return 1 if k > 0 else 0
print(BellMatrix(f, 9))

T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(j, k - j)*<>, where <> denote the second order Eulerian numbers (extending Knuth's notation).
T(n, k) = n!*[z^k][t^n] exp(z*(exp(t) - t - 1)).
T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(n, k - j)*{n - k + j, j}, where {n, k} denotes the Stirling set numbers.
T(n, k) = (n - 1) * T(n-2, k-1) + k * T(n-1, k) with suitable boundary conditions.
T(n + k, k) = A269939(n, k), which might be called the Ward set numbers.

A376544 a(n) is the number of singleton commuting ordered set partitions.

Original entry on oeis.org

1, 1, 2, 8, 40, 242, 1784, 15374, 151008, 1669010, 20503768, 277049126, 4083693200, 65211041690, 1121435565384, 20662801363790, 406100030507200, 8480197575505442, 187500501495191480, 4376026842424336886, 107506303414618515696, 2773174380946415844266

Offset: 0

Raul Penaguiao, Sep 27 2024

nonn

a(n) is also the dimension of the span of chromatic quasi-symmetric invariants of generalized permutahedra.

			a(2) = 2 because the ordered set partitions 1|2 and 2|1 are counted only once.
a(3) = 8, all ordered set partitions with length 3 (e.g. 1|2|3) are counted only once.
a(4) = 40 counts 1|34|2 separately to 2|34|1, but treats 1|2|34 as the same as 2|1|34 since only adjacent singletons can commute.

Alois P. Heinz, Table of n, a(n) for n = 0..435
Vladimir Grujić, Counting faces of graphical zonotopes, Ars Math. Contemp., 13(1), 2017; arXiv preprint, arXiv:1604.06931 [math.CO], 2017.
Raul Penaguiao, The kernel of chromatic quasisymmetric functions on graphs and hypergraphic polytopes, Journal of Combinatorial Theory, Series A, 175, 105258, 2020.

Corresponds to a subset of elements counted in A000670.

Cf. A005840, A113059, A307389, A358623.

b:= proc(n, p) option remember; `if`(n=0, 1/p!, add(
      b(n-j, 0)*binomial(n, j)/p!, j=2..n)+b(n-1, p+1)*n)
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Nov 19 2024

\\ here B(n,k) is A008299 or A358623.
B(n, k) = {sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) ); }
a(n)={sum(k=0, n, binomial(n,k)*sum(j=0, k\2, B(k,j)*j!*(j+1)^(n-k)))} \\ Andrew Howroyd, Sep 27 2024

seq(n)=my(g=exp(x + O(x*x^n))); Vec(serlaplace(g/(1 - g*(g-x-1)))) \\ Andrew Howroyd, Sep 27 2024

Asymptotic growth: a(n) = n! * b^(-n) * c, where b is the unique positive root of exp(2*x) = 1 + e^x + x*e^x, and c is 0.722487... .
From Andrew Howroyd, Sep 27 2024: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} binomial(n,k)*A358623(k,j)*j!*(j+1)^(n-k).
E.g.f.: exp(x)/(1 - exp(x)*(exp(x)-x-1)). (End)
In the notation above, c = 1/(b*(2*exp(b) - b - 2)). - Vaclav Kotesovec, Nov 21 2024

a(10) onwards from Andrew Howroyd, Sep 27 2024

A000504 S2(j,2j+3) where S2(n,k) is a 2-associated Stirling number of the second kind.

Original entry on oeis.org

1, 56, 1918, 56980, 1636635, 47507460, 1422280860, 44346982680, 1446733012725, 49473074851200, 1774073543492250, 66681131440423500, 2624634287988087375, 108060337458000427500, 4647703259223579555000, 208548093035794902390000, 9749651260035434678555625

Offset: 1

N. J. A. Sloane

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.

H. W. Gould, Harris Kwong, Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.
M. Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., 56 (1934), p. 87-95.

Cf. A008299, A000497.

gf := (u,t)->exp(u*(exp(t)-1-t)); S2a := j->simplify(subs(u=0,t=0,diff(gf(u,t),u$j,t$(2*j+3)))/j!); for i from 1 to 20 do S2a(i); od; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000

a[n_] := n (n+1) (10n^2+15n+2) (2n+3)!! / 810; Array[a, 20] (* Jean-François Alcover, Feb 09 2016, after Mark van Hoeij *)

It appears a(n) = 2^(n+1)*GAMMA(n+5/2)*(n^2+n)*(10*n^2+15*n+2)/(405*Pi^(1/2)). - Mark van Hoeij,  Oct 26 2011.
G.f.: x*(7*(5-30*x) * hypergeom([4, 9/2],[],2*x) - 26*hypergeom([3, 7/2],[],2*x))/9. - Mark van Hoeij,  Apr 07 2013
(n-1)*(10*n^2-5*n-3)*a(n) - (2*n+3)*(n+1)*(10*n^2+15*n+2)*a(n-1) = 0. - R. J. Mathar, Jun 09 2018

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000

A171738 Number of n-digit terms in A115853.

Original entry on oeis.org

0, 9, 9, 252, 819, 11754, 72585, 803448, 6978159, 73047510, 744922341, 8023947732, 88219609227, 993117723282, 11397388906305, 132852212160624, 1568346473860839, 18699577205645646, 224600363892164061, 2711096523623447820, 32815659723020049411, 397495008150096639114

Offset: 1

Zak Seidov, Dec 17 2009

Matthew House, Table of n, a(n) for n = 1..996
Index entries for linear recurrences with constant coefficients, signature (220, -23595, 1644214, -83716633, 3320655624, -106839372783, 2866614688938, -65446367283297, 1290934186719996, -22263630922575471, 338916526924263606, -4589394288200909781, 55636105851803049936, -607029027026607279171, 5987676299284026149946, -53597962913874542531511, 436793267723498418791916, -3249590798297591166114241, 22121531283842396531966386, -138066798468560248794910299, 791349936276906779789655496, -4171090447580882317650845341, 20240413531917813060282288558, -90503364704585154511917229755, 373146584463992545811390561028, -1419285864685331116742908160565, 4981481723295791035938228563970, -16135667263673226340044156972087, 48229932136237067328168799757664, -132993220541110610037595712319729, 338163168478885883461275641450094, -792363490141660083452341647812844, 1709441117632380221064502861863864, -3391977966130763237758648740886704, 6182428869770409717691044568257504, -10334792538707263177934951305545664, 15815751917789533624508526223437184, -22110429327513201841008428041853184, 28167208461596745245212310681342464, -32604111806052111009853487965004800, 34175732364859420044995467888502784, -32312338074096507084007090667556864, 27429278013461965544387449830678528, -20791237185298123019559840061734912, 13980827805208637058110391660085248, -8274715534505360843464786947735552, 4269275618057660740303251391905792, -1897199932751339113592691061948416, 715083586820263550683946460119040, -224022790650071250708428803276800, 56733883860143782469881036800000, -11154067599888012976481894400000, 1596567281510638039125196800000, -147945254464527104212992000000, 6658606584104736522240000000).

Cf. A115853 (numbers where every present decimal digit occurs more than once).

Table[9 Sum[k! Binomial[9, k] (-1)^i Binomial[n, i] StirlingS2[n - i, k - i + 1], {k, 0, 9}, {i, 0, Min[n, k + 1]}], {n, 21}] (* Matthew House, Sep 06 2020 *)

From Matthew House, Sep 13 2020: (Start)
a(n) = Sum_{k=0..9} k!*C(9,k)*(S_2(n,k) + k*S_2(n,k+1)) = 9*Sum_{k=0..9} k!*C(9,k)*S_2(n,k+1), where S_2(n,k) = A008299(n,k).
a(n) = 9*Sum_{k=0..10} (-1)^k*9!/(10-k)!*C(n,k)*(10-k)^(n-k) for n >= 10, where 0^0 = 1.
All terms from a(11) onward satisfy a linear recurrence with characteristic polynomial (1-x)^10*(2-x)^9*(3-x)^8*(4-x)^7*(5-x)^6*(6-x)^5*(7-x)^4*(8-x)^3*(9-x)^2*(10-x). (End)

More terms from Matthew House, Sep 06 2020

cp's OEIS Frontend

A280202 Number of topologies on an n-set X such that for all x in X there is a y in X such that x and y are topologically indistinguishable.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A000497 S2(j,2j+2) where S2(n,k) is a 2-associated Stirling number of the second kind.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A259877 If n is even then a(n) = n!/( 2^(n/2)*(n/2)! ), otherwise a(n) = n!/( 3*2^((n-1)/2)*((n-3)/2)! ).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A291104 Number of maximal irredundant sets in the n X n rook graph.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A291543 Array read by antidiagonals: T(m,n) = number of maximal irredundant sets in the lattice (rook) graph K_m X K_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A352607 Triangle read by rows. T(n, k) = Bell(k)*Sum_{j=0..k}(-1)^(k+j)*binomial(n, n-k+j)*Stirling2(n-k+j, j) for n >= 0 and 0 <= k <= floor(n/2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Python

A259877 If n is even then a(n) = n!/( 2^(n/2)(n/2)! ), otherwise a(n) = n!/( 32^((n-1)/2)*((n-3)/2)! ).

A352607 Triangle read by rows. T(n, k) = Bell(k)Sum_{j=0..k}(-1)^(k+j)binomial(n, n-k+j)*Stirling2(n-k+j, j) for n >= 0 and 0 <= k <= floor(n/2).