A082134 -id:A082134 - cp's OEIS frontend

A002697 a(n) = n*4^(n-1).

0, 1, 8, 48, 256, 1280, 6144, 28672, 131072, 589824, 2621440, 11534336, 50331648, 218103808, 939524096, 4026531840, 17179869184, 73014444032, 309237645312, 1305670057984, 5497558138880, 23089744183296

Offset: 0

N. J. A. Sloane

Coefficient of x^(2n-2) in Chebyshev polynomial T(2n) is -a(n).
Let M_n be the n X n matrix m_(i,j) = 1 + 2*abs(i-j); then det(M_n) = (-1)^(n-1)*a(n-1). - Benoit Cloitre, May 28 2002
Number of subsequences 00 in all words of length n+1 on the alphabet {0,1,2,3}. Example: a(2)=8 because we have 000,001,002,003,100,200,300 (the other 57=A125145(3) words of length 3 have no subsequences 00). a(n) = Sum_{k=0..n} k*A128235(n+1, k). - Emeric Deutsch, Feb 27 2007
Let P(A) be the power set of an n-element set A. Then a(n) = the sum of the size of the symmetric difference of x and y for every subset {x,y} of P(A). - Ross La Haye, Dec 30 2007 (See the comment from Bernard Schott below.)
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then remove (y,x) from B when (x,y) is in B and x != y and call this R35. Then a(n) = the sum of the size of the symmetric difference of x and y for every (x,y) of R35. [proposed edit of comment just above; by Ross La Haye]
The numbers in this sequence are the Wiener indices of the graphs of n-cubes (Boolean hypercubes). For example, the 3-cube is the graph of the standard cube whose Wiener index is 48. - K.V.Iyer, Feb 26 2009
From Gary W. Adamson, Sep 06 2009: (Start)
Starting (1, 8, 48, ...) = 4th binomial transform of [1, 4, 0, 0, 0, ...].
Equals the sum of terms in 2^n X 2^n semi-magic square arrays in which each row and column is composed of a binomial frequency of terms in the set (1, 3, 5, 7, ...).
The first few such arrays = [1] [1,3; 3,1]; /Q.
  [1, 3, 5, 3;
   3, 1, 3, 5;
   5, 3, 1, 3;
   3, 5, 3, 1]
(sum of terms = 48, with a binomial frequency of (1, 2, 1) as to (1, 3, 5) in each row and column)
  [1, 3, 5, 3, 5, 7, 5, 3;
   3, 1, 3, 5, 7, 5, 3, 5;
   5, 3, 1, 3, 5, 3, 5, 7;
   3, 5, 3, 1, 3, 5, 7, 5;
   5, 7, 5, 3, 1, 3, 5, 3;
   7, 5, 3, 5, 3, 1, 3, 5;
   5, 3, 5, 7, 5, 3, 1, 3;
   3, 5, 7, 5, 3, 5, 3, 1]
(sum of terms = 256, with each row and column composed of one 1, three 3's, three 5's, and one 7)
... (End)
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the sum of the size of the intersection of x and y for every (x,y) of B. - Ross La Haye, Jan 05 2013
Following the last comment of Ross, A002699 is the similar sequence when "intersection" is replaced by "symmetric difference" and A212698 is the similar sequence when "intersection" is replaced by "union". - Bernard Schott, Jan 04 2013
Also, following the first comment of Ross, A082134 is the similar sequence when "symmetric difference" is replaced by "intersection" and A133224 is the similar sequence when "symmetric difference" is replaced by "union". - Bernard Schott, Jan 15 2013
Let [n] denote the set {1,2,3,...,n} and denote an n-permutation of the elements of [n] by p = p(1)p(2)p(3)...p(n), where p(i) is the i-th entry in the linear order given by p. Then (p(i),p(j)) is an inversion of p if i < j but p(i) > p(j). Denote the number of inversions of p by inv(p) and call a 2n-permutation p = p(1)p(2)...p(2n) 2-ordered if p(1) < p(3) < ... < p(2n-1) and p(2) < p(4) < ... < p(2n). Then Sum(inv(p)) = n*4^(n-1), where the sum is taken over all 2-ordered 2n-permutations of p. See Bona reference below. - Ross La Haye, Jan 21 2014
Sum over all peaks of Dyck paths of semilength n of the product of the x and y coordinates. - Alois P. Heinz, May 29 2015
Sum of the number of all edges over all j-dimensional subcubes of the boolean hypercube graph of dimension n, Q_n, for all j, so a(n) = Sum_{j=1..n} binomial(n,j)*2^(n-j) * j*2^(j-1). - Constantinos Kourouzides, Mar 24 2024

			From _Bernard Schott_, Jan 04 2013: (Start)
See the comment about intersection of X and Y.
If A={b,c}, then in P(A) we have:
{b}Inter{b}={b},
{b}Inter{b,c}={b},
{c}Inter{c}={c},
{c}Inter{b,c}={c},
{b,c}Inter{b}={b},
{b,c}Inter{c}={c},
{b,c}Inter{b,c}={b,c}
and : #{b}+ #{b}+ #{c}+ #{c}+ #{b}+ #{c}+ #{b,c} = 8 = 2*4^(2-1) = a(2).
The other intersections are empty.
(End)

Miklos Bona, Combinatorics of Permutations, Chapman and Hall/CRC, 2004, pp. 1, 43, 64.
C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 516.
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).

Danny Rorabaugh, Table of n, a(n) for n = 0..1000
F. Ellermann, Illustration of binomial transforms
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 414
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Constantinos Kourouzides, A double counting argument on the hypercube graph
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.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Hypercube Graph
Eric Weisstein's World of Mathematics, Wiener Index
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A000051, A000302, A000984, A001792, A002457, A002699, A027656, A038231, A082134, A083672, A125145, A128235, A133224, A212698.

A002697:=1/(4*z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation
A002697:=n->n*4^(n-1): seq(A002697(n), n=0..30); # Wesley Ivan Hurt, Mar 30 2014

Table[n 4^(n - 1), {n, 0, 30}] (* Harvey P. Dale, Jan 18 2012 *)
LinearRecurrence[{8, -16}, {0, 1}, 30] (* Harvey P. Dale, Jan 18 2012 *)
CoefficientList[Series[x/(1 - 4 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

PARI
```
a(n)=if(n<0,0,n*4^(n-1))
```

[n*4^(n-1) for n in range(22)] # Danny Rorabaugh, Mar 27 2015

a(n) = n*4^(n-1).
G.f.: x/(1-4x)^2. a(n+1) is the convolution of powers of 4 (A000302). - Wolfdieter Lang, May 16 2003
Third binomial transform of n. E.g.f.: x*exp(4x). - Paul Barry, Jul 22 2003
a(n) = Sum_{k=0..n} k*binomial(2*n, 2*k). - Benoit Cloitre, Jul 30 2003
For n>=0, a(n+1) = Sum_{i+j+k+l=n} binomial(2i, i)*binomial(2j, j)*binomial(2k, k)*binomial(2l, l). - Philippe Deléham, Jan 22 2004
a(n) = Sum_{k=0..n} 4^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2. - Paul Barry, Oct 15 2004
Sum_{n>0} 1/a(n) = 8*log(2) - 4*log(3). - Jaume Oliver Lafont, Sep 11 2009
a(0) = 0, a(n) = 4*a(n-1) + 4^(n-1). - Vincenzo Librandi, Dec 31 2010
a(n+1) is the convolution of A000984 with A002457. - Rui Duarte, Oct 08 2011
a(0) = 0, a(1) = 1, a(n) = 8*a(n-1) - 16*a(n-2). - Harvey P. Dale, Jan 18 2012
a(n) = A002699(n)/2 = A212698(n)/3. - Bernard Schott, Jan 04 2013
G.f.: W(0)*x/2 , where W(k) = 1 + 1/( 1 - 4*x*(k+2)/( 4*x*(k+2) + (k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(5/4). - Amiram Eldar, Oct 28 2020
a(n) = (1/2)*Sum_{k=0..n} k*binomial(2*n, k). Compare this with the formula of Benoit Cloitre above. - Wolfdieter Lang, Nov 12 2021
a(n) = (-1)^(n-1)*det(M(n)) for n > 0, where M(n) is the n X n symmetric Toeplitz matrix whose first row consists of 1, 3, ..., 2*n-1. - Stefano Spezia, Aug 04 2022

A057711 a(0)=0, a(1)=1, a(n) = n*2^(n-2) for n >= 2.

Original entry on oeis.org

0, 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560, 5632, 12288, 26624, 57344, 122880, 262144, 557056, 1179648, 2490368, 5242880, 11010048, 23068672, 48234496, 100663296, 209715200, 436207616, 905969664, 1879048192, 3892314112, 8053063680

Offset: 0

Bernhard Wolf (wolf(AT)cs.tu-berlin.de), Oct 24 2000

Number of states in the planning domain FERRY, when n-3 cars are at one of two shores while the (n-2)nd car may be on the ferry or at one of the shores.
If the ferry could board any number of cars (instead of only one), the number of states would form the Pisot sequence P(2,6) (A008776). In addition, if k shores existed, the sequence would form the Pisot sequence P(k,k(k+1)). This corresponds to the BRIEFCASE planning domain.
a(i) is the number of occurrences of the number 1 in all palindromic compositions of n = 2*(i+1). - Silvia Heubach (sheubac(AT)calstatela.edu), Jan 10 2003. E.g., there are 5 palindromic compositions of 6, namely 111111 11211 2112 1221 141, containing a total of 16 1's.
Number of occurrences of 00's in all circular binary words of length n. Example: a(3)=6 because in the circular binary words 000, 001, 010, 011, 100, 101, 110 and 111 we have a total of 3+1+1+0+1+0+0+0=6 occurrences of 00. a(n) = Sum_{k=0..n} k*A119458(n,k). - Emeric Deutsch, May 20 2006
a(n) is the number of permutations on [n] for which the entries of each left factor form a circular subinterval of [n]. A subset I of [n] forms a circular subinterval of [n] if it is an ordinary interval [a,b] or has the form [1,a]-union-[b,n] for 1 <= a < b <= n. For example, (5,4,2) is a left factor of the permutation (5,4,2,1,3) which does not form a circular subinterval of [5] and a(4)=16 counts all 24 permutations of [4] except the eight whose first two entries are 1,3 (in either order) or 2,4. - David Callan, Mar 30 2007
a(n) is the total number of runs in all Boolean (n-1)-strings. For example, the 8 Boolean 3-strings, 000, 001, 010, 011, 100, 101, 110, 111 have 1, 2, 3, 2, 2, 3, 2, 1 runs respectively. - David Callan, Jul 22 2008
From Gary W. Adamson, Jul 31 2010: (Start)
Starting with "1" = (1, 2, 4, 8, ...) convolved with (1, 0, 2, 4, 8, ...).
Example: a(6) = 96 = (32, 16, 8, 4, 2, 1) dot (1, 0, 2, 4, 8, 16) = (32 + 0 + 16 + 16 + 16, + 16) = 32 + 4*16 (End)
An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A087447 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Starting with 1 = (1, 1, 2, 4, 8, 16, ...) convolved with (1, 1, 3, 7, 15, 31, ...). - Gary W. Adamson, Oct 26 2010
a(n) is the number of ways to draw simple polygonal chains for n vertices lying on a circle. - Anton Zakharov, Dec 31 2016
Also the number of edges, maximal cliques, and maximum cliques in the n-folded cube graph for n > 3. - Eric W. Weisstein, Dec 01 2017 and Mar 21 2018
Number of pairs of compositions of n corresponding to a seaweed algebra of index n-2 for n > 2. - Nick Mayers, Jun 25 2018
Starting with 1, 2, 6, 16, ..., number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>2, 1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements. - Sergey Kitaev, Dec 08 2020

			a(1)=6 because the palindromic compositions of n=4 are 4, 1+2+1, 1+1+1+1 and 2+2 and they contain 6 ones. - Silvia Heubach (sheubac(AT)calstatela.edu), Jan 10 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
O. Aichholzer, A. Asinowski, and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
A. Burstein, S. Kitaev, and T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.
P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin. 69 (2003), 65-78.
Vincent Coll et al., Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9.1 (2015).
Vladimir Dergachev and Alexandre Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10.2 (2000): 331-343.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
M. Ghallab et al., FERRY domain
M. Ghallab, A. Howe et al., PDDL - The Planning Domain Definition Language, Version 1.2, Technical Report CVC TR-98-003/DCS TR-1165. Yale Center for Computational Vision and Control, 1998.
Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients, J. Int. Seq. (2023) Art. 23.4.3. See p. 12.
Eric Weisstein's World of Mathematics, Folded Cube Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
B. Wolf, Creating state sets
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A082133, A082134, A082135, A082136.
Pisot sequence P(2, 6) (A008776), Pisot sequence P(k, k(k+1))
Cf. A119458.
Cf. A082140, A082141, A082138, A082139, A080951, A080929.

[Ceiling(n*2^(n-2)) : n in [0..40]]; // Vincenzo Librandi, Sep 22 2011

Join[{0, 1}, Table[n 2^(n - 2), {n, 2, 30}]] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{0, 1}, LinearRecurrence[{4, -4}, {2, 6}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x (1 - 2 x + 2 x^2)/(1 - 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n)=ceil(n*2^(n-2)) \\ Charles R Greathouse IV, Oct 31 2011

x='x+O('x^50); concat(0, Vec(x*(1-2*x+2*x^2)/(1-2*x)^2)) \\ Altug Alkan, Nov 01 2015

a(n) = ceiling(n*2^(n-2)).
Binomial transform of (0, 1, 0, 3, 0, 5, 0, 7, ...).
From Paul Barry, Apr 06 2003: (Start)
a(0)=0, a(n) = n*(0^(n-1) + 2^(n-1))/2, n > 0.
a(n) = Sum_{k=0..n} binomial(n, 2k+1)*(2k+1).
E.g.f.: x*exp(x)*cosh(x). (End)
The sequence 1, 1, 6, 16, ... is the binomial transform of A016813 with interpolated zeros. - Paul Barry, Jul 25 2003
For n > 1, a(n) = Sum_{k=0..n} (k-n/2)^2 C(n, k). (n+1)*a(n) = A001788(n). - Mario Catalani (mario.catalani(AT)unito.it), Nov 26 2003
From Paul Barry, May 07 2004: (Start)
a(n) = n*2^(n-2) - Sum_{k=0..n} binomial(n, k)*k*(-1)^k.
G.f.: x*(1-2*x+2*x^2)/(1-2*x)^2. (End)
a(n+1) = ceiling(binomial(n+1,1)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
a(n+1) = Sum_{k=0..n} A196389(n,k)*2^k. - Philippe Deléham, Oct 31 2011
a(0)=0, a(1)=1, a(2)=2, a(3)=6, a(n+1) = 4*a(n)-4*a(n-1) for n >= 3. - Philippe Deléham, Feb 20 2013
a(n) = A002064(n-1) - A002064(n-2), for n >= 2. - Ivan N. Ianakiev, Dec 29 2013
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=1} 1/a(n) = 4*log(2) - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(3/2) - 1. (End)

A082135 Expansion of e.g.f. xexp(4x)*cosh(x).

Original entry on oeis.org

0, 1, 8, 51, 304, 1765, 10104, 57239, 321248, 1787337, 9864040, 54035707, 294031632, 1590368429, 8556082136, 45812239455, 244255416256, 1297362967441, 6867617339592, 36243304518083, 190746485895920, 1001394643462773

Offset: 0

Paul Barry, Apr 06 2003

Binomial transform of A082134. 4th binomial transform of (0,1,0,3,0,5,0,7,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-94,240,-225).

Cf. A082133, A082136.

[n*(3^(n-1)+5^(n-1))/2: n in [0..30]]; // G. C. Greubel, Feb 05 2018

With[{nn = 20}, CoefficientList[Series[x Exp[4*x] Cosh[x], {x, 0, nn}], x] Range[0, nn]!] (* T. D. Noe, Dec 10 2012 *)
Table[n*(3^(n-1)+5^(n-1))/2, {n,0,30}] (* G. C. Greubel, Feb 05 2018 *)
LinearRecurrence[{16,-94,240,-225},{0,1,8,51},40] (* Harvey P. Dale, Sep 13 2024 *)

for(n=0,30, print1(n*(3^(n-1)+5^(n-1))/2, ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = n*(3^(n-1) + 5^(n-1))/2.
E.g.f.: x*exp(4x)*cosh(x).
G.f.: x*(17*x^2-8*x+1) / ((3*x-1)^2*(5*x-1)^2). [Colin Barker, Dec 10 2012]

A082136 Expansion of e.g.f. xexp(5x)*cosh(x).

Original entry on oeis.org

0, 1, 10, 78, 560, 3880, 26400, 177632, 1185280, 7853184, 51699200, 338331136, 2201948160, 14258137088, 91894620160, 589744496640, 3770069811200, 24015941435392, 152494553825280, 965472423378944, 6096346179174400

Offset: 0

Paul Barry, Apr 06 2003

Binomial transform of A082135. 5th binomial transform of (0,1,0,3,0,5,0,7,...)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-148,480,-576).

Cf. A082134.

[n*(4^(n-1)+6^(n-1))/2: n in [0..30]]; // G. C. Greubel, Feb 05 2018

With[{nmax = 50}, CoefficientList[Series[x*Exp[5*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* or *) Table[n*(4^(n-1)+6^(n-1))/2, {n,0,30}] (* G. C. Greubel, Feb 05 2018 *)

for(n=0,30, print1(n*(4^(n-1)+6^(n-1))/2, ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = n*(4^(n-1) + 6^(n-1))/2.
E.g.f.: x*exp(5*x)*cosh(x).
G.f. x*(1-10*x+26*x^2) / ( (6*x-1)^2*(4*x-1)^2 ). - R. J. Mathar, Nov 24 2012

A082133 Expansion of e.g.f. xexp(2x)*cosh(x).

Original entry on oeis.org

0, 1, 4, 15, 56, 205, 732, 2555, 8752, 29529, 98420, 324775, 1062888, 3454373, 11160268, 35872275, 114791264, 365897137, 1162261476, 3680494655, 11622614680, 36611236221, 115063885244, 360882185515, 1129718145936

Offset: 0

Paul Barry, Apr 06 2003

Binomial transform of A057711. 2nd binomial transform of (0,1,0,3,0,5,0,7,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mark Shattuck, Enumeration of consecutive patterns in flattened Catalan words, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 6.
Index entries for linear recurrences with constant coefficients, signature (8,-22,24,-9).

Cf. A082134, A082135, A082136.

List([0..10^2], n->Sum([1..n], k->Sum([1..3], j->Stirling2(n,j)))); # Muniru A Asiru, Feb 06 2018

[n*(1^(n-1) + 3^(n-1))/2: n in [0..30]]; // G. C. Greubel, Feb 05 2018

with (combinat):seq(sum(sum(stirling2(n, j),j=1..3), k=1..n), n=0..24); # Zerinvary Lajos, Dec 04 2007

With[{nn=30},CoefficientList[Series[x Exp[2x]Cosh[x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 30 2012 *)
Table[n*(1^(n-1) + 3^(n-1))/2, {n,0,30}] (* G. C. Greubel, Feb 05 2018 *)
Table[Sum[Sum[StirlingS2[n,j], {j,1,3}], {k,1,n}], {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)

for(n=0,30, print1(n*(1^(n-1) + 3^(n-1))/2, ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = n*(1^(n-1) + 3^(n-1))/2.
E.g.f.: x*exp(2x)*cosh(x).
G.f.: x*(1-4*x+5*x^2) / ( (3*x-1)^2*(x-1)^2 ). - R. J. Mathar, Nov 24 2012
a(n) = Sum_{k=1..n} (Sum_{j=1..3} Stirling2(n,j)). - G. C. Greubel, Feb 07 2018

Definition clarified by Harvey P. Dale, Apr 30 2012

A133224 Let P(A) be the power set of an n-element set A and let B be the Cartesian product of P(A) with itself. Remove (y,x) from B when (x,y) is in B and x <> y and let R35 denote the reduced set B. Then a(n) = the sum of the sizes of the union of x and y for every (x,y) in R35.

Original entry on oeis.org

0, 2, 14, 78, 400, 1960, 9312, 43232, 197120, 885888, 3934720, 17307136, 75509760, 327182336, 1409343488, 6039920640, 25770065920, 109522223104, 463857647616, 1958507577344, 8246342451200

Offset: 0

Ross La Haye, Dec 30 2007, Jan 03 2008

A082134 is the analogous sequence if "union" is replaced by "intersection" and A002697 is the analogous sequence if "union" is replaced by "symmetric difference". Here, X union Y = Y union X are considered as the same Cartesian product [Relation (37): U_Q(n) in document of Ross La Haye in reference], if we want to consider that X Union Y and Y Union X are two distinct Cartesian products, see A212698. [Bernard Schott, Jan 11 2013]

			a(2) = 14 because for P(A) = {{},{1},{2},{1,2}} |{} union {1}| = 1, |{} union {2}| = 1, |{} union {1,2}| = 2, |{1} union {2}| = 2, |{1} union {1,2}| = 2 and |{2} union {1,2}| = 2, |{} union {}| = 0, |{1} union {1}| = 1, |{2} union {2}| = 1, |{1,2} union {1,2}| = 2, which sums to 14.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
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 (12,-52,96,-64).

Cf. A027471, A002697, A082134, A212698.

[n*(2^(n-2) + 3*2^(2*n-3)): n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

LinearRecurrence[{12,-52,96,-64},{0,2,14,78},30] (* Harvey P. Dale, Jan 24 2019 *)

a(n) = n*(2^(n-2) + 3*2^(2*n-3)).
G.f.: 2*x*(7*x^2-5*x+1) / ((2*x-1)^2*(4*x-1)^2). [Colin Barker, Dec 10 2012]
E.g.f.: exp(2*x)*(1 + 3*exp(2*x))*x. - Stefano Spezia, Aug 04 2022

A133789 Let P(A) denote 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 x is not a subset of y and y is not a subset of x, 1) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 2) x and y intersect but for which x is not a subset of y and y is not a subset of x.

Original entry on oeis.org

0, 1, 4, 16, 70, 316, 1414, 6196, 26590, 112156, 466774, 1923076, 7863310, 31972396, 129459334, 522571156, 2104535230, 8460991036, 33972711094, 136277478436, 546270602350, 2188566048076, 8764718254054, 35090241492916, 140455083984670, 562102715143516

Offset: 0

Ross La Haye, Jan 03 2008, Jan 08 2008

nonn

Also, number of even binomial coefficient in rows 0 to 2^n of Pascal's triangle. [Aaron Meyerowitz, Oct 29 2013]

			a(3) = 16 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we see that
{1} and {2},
{1} and {3},
{2} and {3},
{1} and {2,3},
{2} and {1,3},
{3} and {1,2}
are disjoint, while
{} and {1},
{} and {2},
{} and {3},
{} and {1,2},
{} and {1,3},
{} and {2,3},
{} and {1,2,3}
are disjoint and one is a subset of the other and
{1,2} and {1,3},
{1,2} and {2,3},
{1,3} and {2,3}
are intersecting, but neither is a subset of the other.
Also, through row 8 of Pascal's triangle the a(3)=16 even entries are 2 (so a(0)=0 and a(1)=1) then 4,6,4 (so a(2)=4) then 10,10 then  6,20,6 then 8,28,56,70,56,28,8. [_Aaron Meyerowitz_, Oct 29 2013]

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. [_Ross La Haye_, Feb 22 2009]

Cf. A082134, A002697, A020522, A006516, A007582, A000302.

Cf. A000225, A000392, A032263.

a(n) = (1/2)(4^n - 2*3^n + 3*2^n - 2).
O.g.f.: x*(1-6*x+11*x^2)/[(-1+x)*(-1+2*x)*(-1+3*x)*(-1+4*x)]. - R. J. Mathar, Jan 11 2008
a(n) = A084869(n)-1 = A016269(n-2)+2^n-1. - Vladeta Jovovic, Jan 04 2008, corrected by Eric Rowland, May 15 2017
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4). [Aaron Meyerowitz, Oct 29 2013]

Edited by N. J. A. Sloane, Jan 20 2008 to incorporate suggestions from several contributors.

A308700 a(n) = n * 2^(n - 2) * (2^(n - 1) - 1).

Original entry on oeis.org

0, 0, 2, 18, 112, 600, 2976, 14112, 65024, 293760, 1308160, 5761536, 25153536, 109025280, 469704704, 2013143040, 8589672448, 36506664960, 154617643008, 652832538624, 2748773826560, 11544861081600, 48378488553472, 202310091276288, 844424829468672, 3518436999168000

Offset: 0

Stefano Spezia, Jun 17 2019

Given a pseudo-graph P of the set X = {1, 2, ..., n}, defined as a graph represented by the discrete topology on the set X (the power set of X), for n > 0, a(n) is the number of edges of the topological graph arising by deleting loops in P (see Theorem 3.3 in Kozae et al.).

			For n = 3, the set X = {1,2,3},
  the power set 2^X = {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, X} and the pseudo-graph P represented by 2^X has the following edges, here grouped into...
  simple loops:
  {1} --- {1}, {2} --- {2}, {3} --- {3} for a total of 3.
  double loops:
  {1,2} --- {1,2}, {1,3} --- {1,3}, {2,3} --- {2,3} for a total of 6 simple loops.
  triple loop:
  X --- X for a total of 3 simple loops.
  simple edges:
  {1} --- {1,2}, {1} --- {1,3}, {1} --- X, {2} --- {1,2}, {2} --- {2,3}, {2} --- X, {3} --- {1,3}, {3} --- {2,3}, {3} --- X, {1,2} --- {1,3}, {1,2} --- {2,3}, {1,3} --- {2,3} for a total of 12.
  double edges:
  {1,2} --- X, {1,3} --- X, {2,3} --- X for a total of 6 simple edges.
  By deleting the loops in P, there remain a total of a(3) = 12 + 6 = 18 edges for the topological graph arising from P.

A. M. Kozae, A. A. El Atik, A. Elrokh and M. Atef, New types of graphs induced by topological spaces, Journal of Intelligent & Fuzzy Systems, vol. 36, no. 6 (2019), pp. 5125-5134; on Research Gate.
Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

Cf. A000225, A001787, A005843, A006516, A036239, A082134, A171476, A171496.

Cf. A082134 (total number of edges of the pseudo-graph P).

Flat(List([0..25], n->n*2^(n-2)*(2^(n-1)-1)))

Magma
```
[n*2^(n-2)*(2^(n-1)-1): n in [0..25]];
```

a:=n->n*2^(n-2)*(2^(n-1)-1): seq(a(n),n=0..25);

Table[n 2^(n - 2)(2^(n - 1) - 1), {n, 0, 31}]

makelist(n*2^(n-2)*(2^(n-1)-1), n, 0, 25);

PARI
```
a(n)=n*2^(n-2)*(2^(n-1)-1);
```

O.g.f.: -2 * x^2 * (-1 + 3*x)/((-1 + 2*x)^2 * (-1 + 4*x)^2).
E.g.f.: (1/2) * exp(2*x) * (-1 + exp(2*x)) * x.
a(n) = 12 * a(n - 1) - 52*a(n - 2) + 96*a(n - 3) - 64*a(n - 4) for n > 3.
a(n) = n * 2^(n - 2) * (2^(n - 1) - 1).
Lim_{n -> infinity} a(n)/a(n - 1) = 4.
a(n) = A082134(n) - A001787(n).
a(n) = A005843(A001787(n)) * A000225(n - 1).
a(n) = n * A006516(n - 1).
a(n) = n * A171476(n - 2).
a(n) = n * A171496(n - 3).

cp's OEIS Frontend

A002697 a(n) = n*4^(n-1).

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A057711 a(0)=0, a(1)=1, a(n) = n*2^(n-2) for n >= 2.

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A082135 Expansion of e.g.f. x*exp(4*x)*cosh(x).

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A082136 Expansion of e.g.f. x*exp(5*x)*cosh(x).

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A082133 Expansion of e.g.f. x*exp(2*x)*cosh(x).

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A133224 Let P(A) be the power set of an n-element set A and let B be the Cartesian product of P(A) with itself. Remove (y,x) from B when (x,y) is in B and x <> y and let R35 denote the reduced set B. Then a(n) = the sum of the sizes of the union of x and y for every (x,y) in R35.

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A082135 Expansion of e.g.f. xexp(4x)*cosh(x).

A082136 Expansion of e.g.f. xexp(5x)*cosh(x).

A082133 Expansion of e.g.f. xexp(2x)*cosh(x).