A007047 -id:A007047 - cp's OEIS frontend

A328044 Number of chains of binary matrices of order n.

1, 3, 299, 28349043, 21262618727925419, 426789461753903103302333992563, 576797123806621878513443912437627670334052360619, 110627172261659730424051586605958905845740712964061737226074854597705843

Offset: 0

S. R. Kannan, Rajesh Kumar Mohapatra, Oct 03 2019

nonn

For n >= 1, a(n) is the number of chains of n X n (0, 1) matrices.
a(n) is also the number of chains in the power set of n^2 elements.
a(n) is the n^2-th term of A007047.
A chain of binary (crisp or Boolean or logical) matrices of order n can be thought of as a fuzzy matrix of order n.
a(n) is the number of distinct n X n fuzzy matrices.
a(n) is the sum of the n^2-th row of triangle A038719.

Alois P. Heinz, Table of n, a(n) for n = 0..20 (first 11 terms from Rajesh Kumar Mohapatra)
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
V. Murali, Combinatorics of counting finite fuzzy subsets, Fuzzy Sets and Systems, 157(17)(2006), 2403-2411.
V. Murali and B. Makamba, Finite Fuzzy Sets, International Journal of General Systems, Vol. 34 (1) (2005), pp. 61-75.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.

Cf. A000079 (subsets of an n-set), A007047 (chains in power set of an n-set).

Cf. A000290 (squares), A002416 (binary relations on an n-set), A038719 (chains of length k in poset).

# P are the polynomials defined in A007047.
A328044 := n -> 2^(n^2)*subs(x=1/2, P(n^2, x)):
seq(A328044(n), n=0..7); # Peter Luschny, Oct 10 2019

Array[2 PolyLog[-#^2, 1/2] - 1 &, 8, 0] (* Michael De Vlieger, Oct 05 2019, after Jean-François Alcover at A007047 *)
Table[2*PolyLog[-n^2, 1/2] - 1 , {n, 0, 29}]

Let T(n, k) denote the number of chains of binary matrices of order n of length k, T(0, 0) = 1, T(0, k) = 0 for k > 0, thus T(n, k) = A038719(n, k).
a(n) = Sum_{k=0..n^2} T(n, k); a(0) = 1.
a(n) = A007047(n^2) = A007047(A000290(n)).

A330301 Number of chains of binary reflexive matrices of order n.

Original entry on oeis.org

1, 1, 11, 18731, 112366270379, 10710751184977536812459, 45614275176047521934969856784739607851, 19643251901558299817275038399757555422179135786779642874411

Offset: 0

S. R. Kannan, Rajesh Kumar Mohapatra, Jan 01 2020

nonn

Also, the number of chains in the power set of (n^2-n) elements.

a(n) is the number of distinct n X n reflexive fuzzy matrices.

S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, Discrete Math., Vol. 340(5) (2017), pp. 1122-1128.

Alois P. Heinz, Table of n, a(n) for n = 0..21
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
V. Murali, Combinatorics of counting finite fuzzy subsets, Fuzzy Sets Syst., 157(17)(2006), 2403-2411.
V. Murali and B. Makamba, Finite Fuzzy Sets, Int. J. Gen. Syst., Vol. 34 (1) (2005), pp. 61-75.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.
S. Nkonkobe, V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO] Apr 2015.

Cf. A007047, A328044.

# P are the polynomials defined in A007047.
a := n -> 2^(n^2-n)*subs(x=1/2, P(n^2-n, x)):
seq(a(n), n=0..7);

Array[2 PolyLog[-(#^2-#), 1/2] - 1 &, 8, 0]
Table[2*PolyLog[-(n^2-n), 1/2] - 1, {n, 0, 19}]
Table[LerchPhi[1/2, -(n^2-n), 2]/2, {n, 0, 9}]

a(n) = A007047(n^2-n).

A330302 Number of chains of 2-element subsets of {0,1, 2, ..., n} that contain no consecutive integers.

Original entry on oeis.org

1, 1, 3, 51, 18731, 408990251, 921132763911411, 324499299994016295527283, 25190248259800264134073495741338539, 576797123806621878513443912437627670334052360619

Offset: 0

S. R. Kannan, Rajesh Kumar Mohapatra, Jan 01 2020

nonn

For n >= 1, a(n) is the number of chains of binary reflexive symmetric matrices of order n.
The number of chains of strictly upper triangular or strictly lower triangular matrices.
Also, number of chains in power set of (n^2-n)/2 elements.
a(n) is the number of distinct reflexive symmetric fuzzy matrices of order n.

Alois P. Heinz, Table of n, a(n) for n = 0..29
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
V. Murali, Combinatorics of counting finite fuzzy subsets, Fuzzy Sets Syst., 157(17)(2006), 2403-2411.
V. Murali and B. Makamba, Finite Fuzzy Sets, Int. J. Gen. Syst., Vol. 34 (1) (2005), pp. 61-75.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.

Cf. A000079, A161680, A038719, A007047, A328044.

# P are the polynomials defined in A007047.
a:= n -> (m-> 2^m*subs(x=1/2, P(m, x)))(n*(n-1)/2):
seq(a(n), n=0..9);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 4,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> `if`(n<2, 1, b(n*(n-1)/2)-1):
seq(a(n), n=0..10);  # Alois P. Heinz, Feb 11 2020

Array[2 PolyLog[-(#^2-#)/2, 1/2] - 1 &, 10, 0]
Table[2*PolyLog[-(n^2-n)/2, 1/2] - 1, {n, 0, 29}]
Table[LerchPhi[1/2, -(n^2-n)/2, 2]/2, {n, 0, 19}]

a(n) = A007047((n^2-n)/2) = A007047(A161680(n)).

A344037 Expansion of e.g.f.: exp(-2*x) / (2 - exp(x)).

Original entry on oeis.org

1, -1, 3, -1, 27, 119, 1203, 11759, 136587, 1771559, 25562403, 405657119, 7022893947, 131714582999, 2660335750803, 57570797728079, 1328913670528107, 32592691757218439, 846383665814342403, 23200396829831840639, 669421949061096575067, 20281206249626017421879

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

G. C. Greubel, Table of n, a(n) for n = 0..420

Cf. A000670, A007047, A008619, A052841, A330603, A346208.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024

nmax = 21; CoefficientList[Series[Exp[-2 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[HurwitzLerchPhi[1/2, -n, -2]/2, {n, 0, 21}]
a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]

def A344037_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-2*x)/(2-exp(x)) ).egf_to_ogf().list()
A344037_list(40) # G. C. Greubel, Jun 11 2024

a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A000670(k).
a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A008619(k).
a(n) = Sum_{k>=0} (k - 2)^n / 2^(k+1).
a(n) = (-2)^n + Sum_{k=0..n-1} binomial(n,k) * a(k).
a(n) ~ n! / (8 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 15 2021

A107464 Number of fuzzy subgroups of rank 3 cyclic group of order (p^n)qr where p, q and r are three distinct prime.

Original entry on oeis.org

11, 51, 175, 527, 1471, 3903, 9983, 24831, 60415, 144383, 339967, 790527, 1818623, 4145151, 9371647, 21037055, 46923775, 104071167, 229638143, 504365055, 1103101951, 2403336191, 5217714175, 11291066367, 24360517631, 52412022783, 112474456063, 240786604031

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), May 27 2005

It would be good to find a formula for a(n,m,l) or generating function for the number of chains in the lattice of subgroups ( these are the fuzzy subgroups )of the direct sum Z_(p^n) + Z_(q^m) + Z_(r^l) for given 3 distinct prime p,q and r and for integers n,m and l.

			a(5) = (2^6)*(5^2+6*5+6)-1= 3903. This is the number of chains in the lattice of subgroups of the direct sum Z_(p^6)+ Z_q + Z_r for 3 distinct prime p,q and r where Z_i is the group of integers modulo i.

V. Murali, Number of chains in the power set of a set with (n+2) elements, specification n^1 1^2, preprint, 2005.
V. Murali and B. B. Makamba, Fuzzy subgroups of finite Abelian groups III, Rhodes University Preprint, 2005.

Colin Barker, Table of n, a(n) for n = 0..1000
V. Murali, FSRG, Rhodes University.
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A007047, A107392.

Vec((16*x^2-26*x+11)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Jan 15 2015

a(n) = 2^(n+1)*(n^2 + 6n + 6) - 1.

G.f.: (16*x^2-26*x+11) / ((x-1)*(2*x-1)^3). - Colin Barker, Jan 15 2015

Missing a(8) inserted by Colin Barker, Jan 15 2015

A259533 Number of restricted barred preferential arrangements of an n-set having 3 bars in which 3 fixed sections are restricted sections and 1 section is a free section.

Original entry on oeis.org

1, 4, 18, 94, 582, 4294, 37398, 378214, 4366422, 56697574, 817979478, 12981058534, 224732536662, 4214866778854, 85130743747158, 1842265527790054, 42525237455785302, 1042966136232956134, 27084277306054500438, 742412698554626764774, 21421502369955072576342, 648998599988032588957414

Offset: 0

Sithembele Nkonkobe, Jul 02 2015

Also, number of preferential fuzzy subsets of length n+2 where the keychains are of length n+2.
Binomial transform of A007047.
Double binomial transform of A000629.
Conjecture: for fixed k = 1,2,..., the sequence obtained by reducing a(n) modulo k is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 24 the sequence becomes [1, 4, 18, 22, 6, 22, 6, 22, 6, ...] with an apparent period of 2 beginning at a(3). - Peter Bala, Jul 08 2022

Robert Israel, Table of n, a(n) for n = 0..387
V. Murali, Ordered partitions and finite fuzzy sets, Far East J. Math. Sci.(FJMS), 21(2006), 12-132.
S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO], 2015.

Cf. A000670, A000629, A007047.

S:= series(exp(3*x)/(2-exp(x)),x,31):
seq(coeff(S,x,j)*j!, j=0..30); # Robert Israel, Aug 11 2015

Range[0, 25]! CoefficientList[Series[E^(3 x)/(2 - E^(x)), {x, 0, 25}], x] (* Vincenzo Librandi, Jul 06 2015 *)

{ my(x = xx + O(xx^40)); Vec(serlaplace(exp(3*x)/(2-exp(x)))) } \\ Michel Marcus, Jul 06 2015

E.g.f.: exp(3*x)/(2-exp(x)).
a(n) = 3^n + Sum_{k = 0..n-1} binomial(n,k)*a(k). - Robert Israel, Aug 11 2015
a(n) ~ 4*n! / (log(2))^(n+1). - Vaclav Kotesovec, Sep 27 2017
a(n) = Sum_{k>=0} (k + 3)^n / 2^(k+1). - Ilya Gutkovskiy, Jun 27 2020
a(n) = 8*A000670(n) - (2^n + 2 + 4*0^n). - Seiichi Manyama, Dec 21 2023

More terms from Michel Marcus, Jul 06 2015

A162312 Triangular array, inverse of 2*P - I, where P is Pascal's triangle and I is the identity matrix.

Original entry on oeis.org

1, -2, 1, 6, -4, 1, -26, 18, -6, 1, 150, -104, 36, -8, 1, -1082, 750, -260, 60, -10, 1, 9366, -6492, 2250, -520, 90, -12, 1, -94586, 65562, -22722, 5250, -910, 126, -14, 1, 1091670, -756688, 262248, -60592, 10500, -1456, 168, -16, 1, -14174522, 9825030

Offset: 0

Peter Bala, Jul 01 2009

We make a few remarks about the general array M(a) := (I - a*P)^-1, where a <> 1, and its connection with weighted sums of powers of positive integers. The present case corresponds to -M(2).
The array M(a) begins
/
| 1/(1-a)
| a/(1-a)^2............... 1/(1-a)
| (a+a^2)/(1-a)^3......... 2*a/(1-a)^2........ 1/(1-a)
| (a+4*a^2+a^3)/(1-a)^4... 3*(a+a^2)/(1-a)^3.. 3*a/(1-a)^2... 1/(1-a)
| ...
\
In the first column the numerator polynomials are the Eulerian polynomials A_n(a). See A008292.
The e.g.f. for this array is
(1)... exp(x*t)/(1-a*exp(t)) = 1/(1-a) + [a/(1-a)^2 + x/(1-a)]*t
+ [(a+a^2)/(1-a)^3 + 2*a*x/(1-a)^2 + x^2/(1-a)]*t^2/2! + ....
The row generating polynomials P_m(x) of the array M(a), which, of course, depend on a, have properties similar to those of the Bernoulli polynomials. They form an Appell sequence and may be expressed in terms of the Eulerian polynomials as
(2)... P_m(x) = sum {k=0..m} binomial(m,k) * A_k(a) / (1-a)^(k+1) * x^(m-k).
As a Newton series we have
(3)... P_m(x) = 1/(1-a)*sum {j = 0..m} sum {k = j..m}(a/(1-a))^j * k! * Stirling2(m,k) * binomial(x,k-j).
The proof of this result in the particular case a = -1 given in [Roman, p. 100] can be easily generalized to a proof of (3).
A result equivalent to (3) is
(4)... P_m(x) = 1/(1-a)*sum {j = 0..m} sum {k = 0..j} (a/(1-a))^j * (-1)^(j-k) * comb(j,k) * (x + k)^m,
which in turn leads to the infinite series expansion
(5)... P_m(x) = sum {k = 0..inf} a^k * (x + k)^m,
provided |a| < 1. See [Nelsen].
The polynomials P_m(x) satisfy the difference equation
(6)... P_m(x) - a*P_m(x + 1) = x^m (recall a <> 1),
which leads easily to the evaluation of the weighted sums of powers of integers
(7)... sum {k = 0..n-1} a^k * k^m = P_m(0) - a^n * P_m(n).
for m = 0,1,2,... and a <> 1.
More generally we have
(8)... sum {k = 0..n-1} a^k * (x + k)^m = P_m(x) - a^n * P_m(x + n).
for m = 0,1,2,... and a <> 1.
In the remaining case a = 1 the sums are evaluated in terms of the Bernoulli polynomials.
The most well-studied case is when a = -1. The row polynomials of the array M(-1) are then one half of the Euler polynomials E_m(x), which may be used to evaluate the alternating sums of powers of integers
(9)... 2*sum {k = 1..n-1} (-1)^k * k^m = E_m(0) - (-1)^n * E_m(n).

			Triangle begins
====================================================
n\k|.....0......1......2......3......4......5......6
====================================================
0..|.....1
1..|....-2......1
2..|.....6.....-4......1
3..|...-26.....18.....-6......1
4..|...150...-104.....36.....-8......1
5..|.-1082....750...-260.....60....-10......1
6..|..9366..-6492...2250...-520.....90....-12......1
...

S. Roman, The Umbral Calculus, Dover Publications.

G. F. C. De Bruyn, Formulas for a + a^2*2^p + a^3*3^p + ... + a^n*n^p, Fibonacci Quart. 33 (1995), no. 2, 98-103.
R. B. Nelsen, Problem E3062, Amer. Math. Monthly, Vol. 94, No. 4 (Apr., 1987), 376-377.

Cf. A000629, A000670, A007047, A008292, A080253, A154921.

#A162312
with(combinat):
T := (n,k) -> (-1)^(n+k)*binomial(n,k)
*add( j!*stirling2(n-k+1,j+1),j = 0..n):
for n from 0 to 9 do
seq(T(n,k), k = 0..n);
end do;

Table[(-1)^(n+k) Binomial[n, k] PolyLog[k-n, 1/2], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2019 *)

matrix(10, 10, n, k, 2*binomial(n-1,k-1) - (n==k))^(-1) \\ Michel Marcus, Jul 12 2018

TABLE ENTRIES
(1)... T(n,k) = (-1)^(n+k) * binomial(n,k) * A000629(n-k).
(2)... T(n,k) = (-1)^(n+k) * binomial(n,k) * sum {j = 0..n} j! * Stirling2(n-k+1,j+1).
GENERATING FUNCTION
(3)... exp(x*t)/(2*exp(t)-1) = 1 + (-2+x)*t + (6-4*x+x^2)*t^2/2!
+ ....
PROPERTIES OF ROW POLYNOMIALS
The row generating polynomials R_n(x) form an Appell sequence. The first few values are R_0(x) = 1, R_1(x) = x-2, R_2(x) = x^2-4*x+6 and R_3(x) = x^3-6*x^2+18*x-26.
They may be recursively computed by means of
(4)... R_n(x) = x^n - 2*sum {k = 0..n-1} binomial(n,k) * R_k(x).
Explicit formulas are
(5)... R_n(x) = sum {j = 0..n} sum {k = j..n} (-2)^j * k! * Stirling2(n,k) * binomial(x,k-j),
(6)... R_n(x) = (-1)^n * sum {j = 0..n} sum {k = j..n} k! * Stirling2(n,k) * binomial(-x+1,k-j),
and
(7)... R_n(x) = sum {j = 0..n} sum {k = 0..j} 2^j * (-1)^k * comb(j,k) * (x + k)^n.
Other expansions include
(8)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1)^k * A000670(k) * (x-1)^(n-k),
(9)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1/2)^k * A080253(k) * (x-1/2)^(n-k)
and
(10)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1)^k * A007047(k) * (x+1)^(n-k).
SUMS OF POWERS OF INTEGERS
The row polynomials satisfy the difference equation
(11)... 2*R_n(x+1) - R_n(x) = x^n,
and so may be used to evaluate the weighted sum of powers of integers
(12)... sum {k = 0..n-1} 2^k * k^m = 2^n*R_m(n) - R_m(0).
For example, m = 3 gives
(13)... sum {k = 0..n-1} 2^k * k^3 = 2^n*(n^3-6*n^2+18*n-26) + 26.
More generally we have
(14)... sum {k = 0..n-1} 2^k * (x + k)^m = 2^n * R_m(x + n) - R_m(x).
RELATIONS WITH OTHER SEQUENCES
(15)... Row sums [1,-1,3,-13,75,...] = (-1)^n*A000670(n).
(16)... Alt. row sums [1,-3,11,-51,299,...] = (-1)^n * A007047(n).
(17)... Column 0: (-1)^n * A000629(n).
(18)... (-2)^n * R_n(1/2) = A080253(n).
(19)... R_n(1-x) = (-1)^n * P_n(x),
where P_n(x) are the row generating polynomials of A154921.
This provides the connection between (12) and the result
(20)... sum {k = 0..n-1} (1/2)^k * k^m = 2*P_m(0) - (1/2)^(n-1) * P_m(n).

Typo corrected by Peter Bala, Nov 05 2010

A107953 Number of chains in the power set lattice of an (n+3)-element set X_(n+3) of specification n^1 2^1 1, that is, n identical objects of one kind, 2 identical objects of another kind and one other kind. It is the same as the number of fuzzy subsets X_(n+3).

Original entry on oeis.org

31, 175, 703, 2415, 7551, 22143, 61951, 167167, 438271, 1122303, 2818047, 6959103, 16941055, 40730623, 96862207, 228130815, 532676607, 1234173951, 2839543807, 6491734015, 14755561471, 33361494015, 75061264383, 168124481535, 375004332031, 833223655423

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), May 28 2005

This sequence is one of a triple sequence A(n,m,l) of the number of fuzzy subsets of a set with n+m+l objects of 3 kinds. There are n,m and l objects for each kind respectively. Here a(n)= A(n,2,1). The sequence A107464 is one other example of A(n,m,l) for m=l=1.

			a(3) = 2^4*((9/2)*16 + 21*3 + 16) - 1 = 2415 which is the number of distinct chains in the power set lattice (or fuzzy subsets) of a set X_(n+3) with 3 kinds of objects, n of one kind, 2 of another and one of yet another.

V. Murali, On the number of fuzzy subsets of an (n+3)-element set of specification n^1 2^1 1, Rhodes University Preprint, 2005.

Colin Barker, Table of n, a(n) for n = 0..1000
Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Cf. A107464, A007047, A107392.

Table[2^(n+1) (n^2/2 (n+13)+21n+16)-1,{n,0,30}] (* or *) LinearRecurrence[ {9,-32,56,-48,16},{31,175,703,2415,7551},30] (* Harvey P. Dale, Feb 10 2015 *)

Vec((48*x^3-120*x^2+104*x-31)/((x-1)*(2*x-1)^4) + O(x^100)) \\ Colin Barker, Jan 15 2015

a(n) = 2^(n+1) * ((n^2/2)*(n+13) + 21*n + 16) - 1.
G.f.: (48*x^3-120*x^2+104*x-31) / ((x-1)*(2*x-1)^4). - Colin Barker, Jan 15 2015
a(0)=31, a(1)=175, a(2)=703, a(3)=2415, a(4)=7551, a(n)=9*a(n-1)- 32*a(n-2)+ 56*a(n-3)-48*a(n-4)+16*a(n-5). - Harvey P. Dale, Feb 10 2015

a(5) corrected Jun 01 2005

Incorrect term deleted by Colin Barker, Jan 15 2015

cp's OEIS Frontend

A328044 Number of chains of binary matrices of order n.

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A330301 Number of chains of binary reflexive matrices of order n.

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A330302 Number of chains of 2-element subsets of {0,1, 2, ..., n} that contain no consecutive integers.

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A344037 Expansion of e.g.f.: exp(-2*x) / (2 - exp(x)).

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A107464 Number of fuzzy subgroups of rank 3 cyclic group of order (p^n)*q*r where p, q and r are three distinct prime.

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A259533 Number of restricted barred preferential arrangements of an n-set having 3 bars in which 3 fixed sections are restricted sections and 1 section is a free section.

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A162312 Triangular array, inverse of 2*P - I, where P is Pascal's triangle and I is the identity matrix.

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A107464 Number of fuzzy subgroups of rank 3 cyclic group of order (p^n)qr where p, q and r are three distinct prime.