A000392 -id:A000392 - cp's OEIS frontend

A294032 Triangle read by rows, T(n, k) = Pochhammer(3, k)*Stirling2(3 + n, 3 + k) for n >= 0 and 0 <= k <= n.

1, 6, 3, 25, 30, 12, 90, 195, 180, 60, 301, 1050, 1680, 1260, 360, 966, 5103, 12600, 15960, 10080, 2520, 3025, 23310, 83412, 158760, 166320, 90720, 20160, 9330, 102315, 510300, 1369620, 2116800, 1890000, 907200, 181440

Offset: 0

Peter Luschny, Oct 22 2017

			Triangle starts:
[0]    1
[1]    6,      3
[2]   25,     30,     12
[3]   90,    195,    180,      60
[4]  301,   1050,   1680,    1260,     360
[5]  966,   5103,  12600,   15960,   10080,    2520
[6] 3025,  23310,  83412,  158760,  166320,   90720,  20160
[7] 9330, 102315, 510300, 1369620, 2116800, 1890000, 907200, 181440

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T(n, 0) = A000392(n+3), T(n, n) = A001710(n+2).
Row sums A002051(n+3), alternating row sums A000225(n+1).
Cf. A028246 (m=1), A053440 (m=2), this seq. (m=3), A293617 (hub).

A294032 := (n, k) -> pochhammer(3, k)*Stirling2(n + 3, k + 3):
seq(seq(A294032(n, k), k=0..n), n=0..7);
T := (n, k) -> A293617(3, n, k): seq(seq(T(n, k), k=0..n), n=0..7);

Table[Pochhammer[3, k] StirlingS2[3 + n, 3 + k], {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 22 2017 *)

for(n=0,10, for(k=0,n, print1((k+2)!*stirling(n+3,k+3,2)/2, ", "))) \\ G. C. Greubel, Nov 19 2017

E.g.f.: (1/2)*exp(x)*(2*y + 9*exp(2*x) + y^2+1-11*exp(3*x)*y + 15*y^2*exp(2*x) - 7*y^2*exp(x) - 13*y^2*exp(3*x) + 4*exp(4*x)*y^2 - 8*exp(x) + 24*y*exp(2*x) - 15*y*exp(x))/(1 - y*(exp(x) - 1))^3.

T(n, k) = A293617(3, n, k).

A341617 Repair factors for Stirling numbers of the second kind.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 30, 210, 840, 2520, 1260, 13860, 13860, 180180, 90090, 30030, 240240, 4084080, 6126120, 116396280, 58198140, 58198140

Offset: 1

Thomas Ward, Feb 16 2021

a(k) is the smallest number with the property that the sequence (a(k)*S(n+k-1, k))_{n>=1}, viewed as a sequence in n, counts the periodic points of some map. Here S(n, k) denotes the Stirling numbers of the second kind, so S(n, k) is the number of ways to partition a set of n elements into k nonempty subsets.
Because these numbers relate to properties of the Stirling number sequence S(n, k) viewed as a sequence in n, it is natural to think of these values as indexed by k.
The values generated by a finite calculation are inherently empirical, calculated as the least common multiple of the first 3000 terms and then checked for the repair property for period up to 50,000. For smaller values of k (those listed here) various ad hoc arguments can be used to check the empirical candidate values.
The empirical values can be calculated using a simple PARI code, and this only gives a possible candidate value for a(k). If that value happens to be (k-1)! then it is correct. If it is a strict divisor of (k-1)! then further checks are needed for each prime in the quotient (candidate value)/(k-1)!. The potentially unbounded nature of the calculation required would make a proof of a closed formula particularly interesting.
(EMPIRICAL PARI CODE TO GENERATE CANDIDATE VALUES) for(k=1, 40, a=vector(3000, n, stirling(n+k-1, k, 2)); b=vector(length(a), n, (1/n)*sumdiv(n, d, moebius(n/d)*a[d])); print1(denominator(b)", "))

			The statement that a(3) = 2 means that the sequence of Stirling numbers S_3 = (1, 6, 25, 90, ...) (that is, the sequence A000392 with an offset of 3) does not have the property of counting periodic points for some map, but does have this property after multiplication by 2 (which gives A028243 with an offset of 2), and 2 is the smallest integer with this property. This specific value is immediately known to be exact, because a(3) divides (3-1)! = 2.

P. Miska and T. Ward, Stirling numbers and periodic points, arXiv:2102.07561 [math.NT], 2021.

Cf. A000392, A028243, A008277, A341991.

It is known that a(k) divides (k-1)!.

There is an explicit formula for a(k), but it involves an in principle infinite calculation as follows: compute the set of rational numbers {(1/n) Sum_{d|n} mu(n/d)*S(d+k-1, k): n>=1}, and then define a(k) to be the least common multiple of the denominators of that (possibly infinite) set of rational numbers. Here mu denotes the classical Möbius function, and the sum is taken over the divisors d of n. Strictly speaking, any calculation only gives candidate values which are initially only known to be a factor of the real value, which in turn is a factor of (k-1)!. For small values of k listed above ad hoc arguments are needed to check the candidate values evaluated as the least common multiple of the denominators of the first 3000 terms.

A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.

Original entry on oeis.org

1, 10, 75, 520, 3556, 24626, 174805, 1279240, 9677151, 75750752, 613656836, 5142797660, 44557627661, 398786697398, 3683575764083, 35084121263136, 344242894197456, 3476490965903174, 36104281709286841, 385257741260565844, 4220537246457019687, 47432055430482106880

Offset: 3

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000292, A000392, A003128, A005493, A049020, A189924, A346843, A346844.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 3), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=3..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^3/3!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
Table[Sum[StirlingS2[n, k] Binomial[k, 3], {k, 0, n}], {n, 3, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 3] BellB[n - k], {k, 0, n}], {n, 3, 24}]
Table[(BellB[n+3] - 6*BellB[n+2] + 8*BellB[n+1] - BellB[n])/6, {n, 3, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^3/3!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,3).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,3) * Bell(n-k).
a(n) = (Bell(n+3) - 6*Bell(n+2) + 8*Bell(n+1) - Bell(n))/6. - Vaclav Kotesovec, Aug 06 2021
a(n) ~ exp(-1 - n + n/LambertW(n)) * (n - LambertW(n))^3 * n^n / (6 * sqrt(1 + LambertW(n)) * LambertW(n)^(n+3)). - Vaclav Kotesovec, Jun 28 2022

A384988 a(n) = Stirling2(n,2)^2 + Stirling2(n,3).

Original entry on oeis.org

0, 1, 10, 55, 250, 1051, 4270, 17095, 68050, 270451, 1075030, 4276735, 17030650, 67881451, 270777790, 1080817975, 4316294050, 17244046051, 68912400550, 275457464815, 1101251874250, 4403270396251, 17607863991310, 70415790601255, 281616141147250, 1126323450484051

Offset: 1

Julian Allagan, Jun 14 2025

Also, one third of the number of proper vertex colorings of the n-complete tripartite graph using exactly 5 interchangeable colors.

The complete 3-partite graph K(n,n,n) has 3n vertices partitioned into three sets of size n each, with edges between every pair of vertices from different sets. 3*a(n) = 0 for n < 2 because we need at least 2 vertices per partition to create 5 nonempty independent sets.

			3*a(2) = 3 because K(2,2,2) can be partitioned into 5 nonempty independent sets in exactly 3 ways.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Richard P. Stanley, Enumerative Combinatorics, Cambridge University Press.
Eric Weisstein's World of Mathematics, Complete Multipartite Graph.
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000392, A000453, A060867, A385432.

[(6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4: n in [1..30]]; // Vincenzo Librandi, Jul 24 2025

Table[(StirlingS2[n, 3] + StirlingS2[n, 2]^2), {n, 1, 20}]

3*a(n) = 2^(2*n - 2) + (1/2)*3^(n - 1) - 3*2^(n - 1) + 3/2 for n >= 1.
G.f.: 1/(4*(1 - 4*x)) + 1/(6*(1 - 3*x)) - 3/(2*(1 - 2*x)) + 3/(2*(1 - x)).
a(n) = A385432(n, 5) / 3 = A060867(n-1) + A000392(n).
From Stefano Spezia, Jun 14 2025: (Start)
a(n) = (6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4.
E.g.f.: (exp(x) - 1)^2*(3*exp(2*x) + 8*exp(x) - 5)/12. (End)
a(n) = A000453(n+2) -10*A000453(n). - R. J. Mathar, Jul 20 2025

A057668 Number of minimal 7-covers of a labeled n-set.

Original entry on oeis.org

1, 988, 549102, 226064280, 76785889587, 22762819040676, 6092115565691584, 1505097773271664000, 348617485585838373333, 76564317282173987801964, 16080209472530744351164146, 3250906483045575317042337960, 635954979082842132795003641239

Offset: 7

Vladeta Jovovic, Oct 16 2000

Eric Weisstein's World of Mathematics, Minimal cover

Cf. A000392, A003468, A016111, A046166-A046169, A005783-A005786, A055066 (unlabeled case).

a(n) = (1/7!) * (127^n - 7 * 126^n + 21 * 125^n - 35 * 124^n + 35 * 123^n - 21 * 122^n + 7 * 121^n - 120^n).

G.f.: x^7 / ((120*x-1)*(121*x-1)*(122*x-1)*(123*x-1)*(124*x-1)*(125*x-1)*(126*x-1)*(127*x-1)). - Colin Barker, Jul 11 2013

Additional term from Colin Barker, Jul 11 2013

A123212 Let S(1) = {1} and, for n > 1, let S(n) be the smallest set containing x, 2x and x^2 for each element x in S(n-1). a(n) is the sum of the elements in S(n).

Original entry on oeis.org

1, 3, 7, 31, 383, 71679, 4313284607, 18447026747376402431, 340282367000167840050178713574329810943, 115792089237316195429848086745536112650120661123018741407845920610578123980799

Offset: 1

John W. Layman, Oct 05 2006

nonn

If we take the cardinality of the set S(n) instead of the sum, we get the Fibonacci numbers 1,2,3,5,8,13,21,34,... If the set mapping uses x -> x, 2x and 3x instead of x -> x, 2x, and x^2, the corresponding sequence consists of the Stirling numbers of the second kind: 1, 6, 25, 90, 301, 966, 3025, ... (A000392).

			Under the indicated set mapping we have {1} -> {1,2} -> {1,2,4} -> {1,2,4,8,16}, giving the sums a(1)=1, a(2)=3, a(3)=7, a(4)=31, etc.

Alois P. Heinz, Table of n, a(n) for n = 1..13

Cf. A000045, A000392, A122554.

s:= proc(n) option remember; `if`(n=1, 1,
      map(x-> [x, 2*x, x^2][], {s(n-1)})[])
    end:
a:= n-> add(i, i=s(n)):
seq(a(n), n=1..10);  # Alois P. Heinz, Jan 12 2022

S[n_] := S[n] = If[n == 1, {1}, {#, 2#, #^2}& /@ S[n-1] // Flatten // Union];
a[n_] := S[n] // Total;
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Apr 22 2022 *)

from itertools import chain, islice
def A123212_gen(): # generator of terms
    s = {1}
    while True:
        yield sum(s)
        s = set(chain.from_iterable((x,2*x,x**2) for x in s))
A123212_list = list(islice(A123212_gen(),10)) # Chai Wah Wu, Jan 12 2022

A129839 a(n) = Stirling_2(n,3)^2.

Original entry on oeis.org

0, 0, 0, 1, 36, 625, 8100, 90601, 933156, 9150625, 87048900, 812307001, 7486748676, 68447640625, 622473660900, 5641104760201, 51003678922596, 460438253730625, 4152386009780100, 37422167780506201, 337103845136750916, 3035761307578140625, 27332814735512302500

Offset: 0

N. J. A. Sloane, Feb 08 2008

H. S. Wilf, A lot of toast, with a side order of roast, manuscript, Jan 04 2002.
Index entries for linear recurrences with constant coefficients, signature (25,-239,1115,-2664,3060,-1296).

Cf. A000392.

StirlingS2[Range[0,30],3]^2 (* Harvey P. Dale, Jan 03 2013 *)

a(n)=(3^n-3<Charles R Greathouse IV, Jan 03 2013

[stirling_number2(n,3)^2for n in range(0,23)] # Zerinvary Lajos, Mar 14 2009

G.f.: x^3*(1+11*x-36*x^2-36*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-6*x)*(1-9*x)).

a(n) = (3^n - 3*2^n + 3)^2/36 for n>0. - Charles R Greathouse IV, Jan 03 2013

Definition corrected (exponent changed from 3 to 2) by Harvey P. Dale, Jan 03 2013

A134165 Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 3) x = y.

Original entry on oeis.org

1, 3, 8, 24, 86, 348, 1478, 6324, 26846, 112668, 467798, 1925124, 7867406, 31980588, 129475718, 522603924, 2104600766, 8461122108, 33972973238, 136278002724, 546271650926

Offset: 0

Ross La Haye, Jan 12 2008

			a(2) = 8 because for P(A) = {{},{1},{2},{1,2}} we have for case 0 {{},{1}}, {{},{2}}, {{},{1,2}} and we have for case 1 {{1},{2}} and we have for case 3 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 2.

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000225, A000392, A032263, A000079.

LinearRecurrence[{10,-35,50,-24},{1,3,8,24},30] (* Harvey P. Dale, Feb 29 2020 *)

a(n) = (1/2)(4^n - 2*3^n + 5*2^n - 2) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + 2*StirlingS2(n+1,2) + 1.

G.f.: (1-7*x+13*x^2-x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). [Colin Barker, Jul 30 2012]

A144521 Tetrahedral numbers k(k+1)(k+2)/6 such that exactly one of k, k+1, and k+2 is prime.

Original entry on oeis.org

0, 20, 56, 84, 165, 220, 364, 455, 680, 816, 1140, 1330, 1771, 2024, 2300, 3654, 4060, 4960, 5456, 7770, 8436, 9139, 10660, 11480, 13244, 14190, 16215, 17296, 18424, 23426, 24804, 26235, 32509, 34220, 37820, 39711, 47905, 50116, 52394, 57155

Offset: 1

Juri-Stepan Gerasimov, Dec 15 2008

nonn

			k=0: Of the three numbers (0,1,2), exactly one is prime, so 0*1*2/6 = 0 is in the sequence.
k=1: Of the three numbers (1,2,3), exactly two are prime, so 1*2*3/6 = 1 is not in the sequence.
k=4: Of the three numbers (4,5,6), exactly one is prime, so 4*5*6/6 = 20 is in the sequence.

Cf. A000040, A000392, A141468, A152622, A152916.

isPr := proc(n) if isprime(n) then 1; else 0; end if; end proc: for n from 0 to 300 do if isPr(n)+isPr(n+1)+isPr(n+2) = 1 then printf("%d,",n*(n+1)*(n+2)/6 ) ; end if; end do: # R. J. Mathar, May 01 2010

Corrected (455, 14190, 17296 inserted, 16560 removed etc.) by R. J. Mathar, May 01 2010

Name and Example section clarified by Jon E. Schoenfield, Aug 06 2017

A144523 Triangular numbers n*(n+1)/2 with n and n+1 composite, where number of prime factors in n > number of prime factors in n+1.

Original entry on oeis.org

36, 210, 300, 528, 1035, 1176, 1275, 1485, 1596, 2080, 2346, 2926, 3240, 3321, 3570, 4095, 4278, 5460, 5565, 6105, 6555, 6903, 7260, 8256, 8778, 9870, 10440, 11628, 11935, 12880, 13695, 14196, 15576, 16653, 17020, 17391, 17955, 20100, 20910, 21736, 22578, 23436, 24310, 25200, 25425

Offset: 1

Juri-Stepan Gerasimov, Dec 15 2008

nonn

Subsequence of A144291 - R. J. Mathar, Jan 17 2009

Prime factors counted with multiplicity. - Harvey P. Dale, Aug 23 2020

			If n=8=2*2*2(number of prime factors = 3) and n+1=9=3*3(number of prime factors = 2), then 8*9/2=36=a(1). If n=20=2*2*5(number of prime factors = 3) and n+1=21=3*7(number of prime factors = 2), then 20*21/2=210=a(2). If n=24=2*2*2*3(number of prime factors = 4) and n+1=25=5*5(number of prime factors = 2), then 24*25/2=300=a(3), etc.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000392, A141468, A152622.

(Times@@#)/2&/@Select[Partition[Range[250],2,1],AllTrue[ #,CompositeQ] && PrimeOmega[#[[1]]]>PrimeOmega[#[[2]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 23 2020 *)

Corrected definition. 2926 inserted and extended. - R. J. Mathar, Jan 17 2009

cp's OEIS Frontend

A294032 Triangle read by rows, T(n, k) = Pochhammer(3, k)*Stirling2(3 + n, 3 + k) for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A341617 Repair factors for Stirling numbers of the second kind.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A384988 a(n) = Stirling2(n,2)^2 + Stirling2(n,3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A057668 Number of minimal 7-covers of a labeled n-set.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A123212 Let S(1) = {1} and, for n > 1, let S(n) be the smallest set containing x, 2x and x^2 for each element x in S(n-1). a(n) is the sum of the elements in S(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A129839 a(n) = Stirling_2(n,3)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

Extensions

Views

Author

Keywords

Examples

A144521 Tetrahedral numbers k(k+1)(k+2)/6 such that exactly one of k, k+1, and k+2 is prime.