A212698 -id:A212698 - 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

A212704 a(n) = 9n10^(n-1).

Original entry on oeis.org

9, 180, 2700, 36000, 450000, 5400000, 63000000, 720000000, 8100000000, 90000000000, 990000000000, 10800000000000, 117000000000000, 1260000000000000, 13500000000000000, 144000000000000000, 1530000000000000000, 16200000000000000000, 171000000000000000000, 1800000000000000000000

Offset: 1

Stanislav Sykora, May 25 2012

Main transitions in systems of n particles with spin 9/2.
Please, refer to the general explanation in A212697.
This particular sequence is obtained for base b=10, corresponding to spin S = (b-1)/2 = 9/2.
Number of 0 needed to write all numbers of n+1 digits. - Bruno Berselli, Jun 30 2014
Essentially the same as A113119. - Bernard Schott, Nov 15 2022
From Bernard Schott, Nov 22 2022: (Start)
Number of nonzero digits needed to write all integers from 1 up to 10^n - 1.
a(n) is a square iff n in { A016754 union A033583\{0} } (see formulas). (End)

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A008591, A011557, A016754, A033583, A033713, A033714, A053541, A081045, A113119.

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212702, A212703 (for b=2..9).

Rest@ CoefficientList[Series[9 x/(10 x - 1)^2, {x, 0, 18}], x] (* or *)
Array[9 # 10^(# - 1) &, 18] (* Michael De Vlieger, Nov 18 2019 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
a(n) = mtrans(n, 10);

def a(n): return 9*n*10**(n-1)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Nov 14 2022

a(n) = n*(b-1)*b^(n-1) with b=10.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 9*x/(10*x-1)^2.
a(n) = 9*A053541(n). (End)
From Bernard Schott, Nov 14 2022: (Start)
a(n+1) - a(n) = 9*A081045(n).
a(n) = A113119(n) for n > 1.
a(n) = A033713(n+1) - A033713(n) = A033714(n+1) - A033714(n).
a(A016754(n)) = (3 * (2n+1) * 10^(2*n*(n+1)))^2.
a(A033583(n)) = (3 * n * 10^(5*n^2))^2. (End)
From Elmo R. Oliveira, May 13 2025: (Start)
E.g.f.: 9*x*exp(10*x).
a(n) = A008591(n)*A011557(n-1).
a(n) = 20*a(n-1) - 100*a(n-2) for n > 2. (End)

A212697 a(n) = 2n3^(n-1).

Original entry on oeis.org

2, 12, 54, 216, 810, 2916, 10206, 34992, 118098, 393660, 1299078, 4251528, 13817466, 44641044, 143489070, 459165024, 1463588514, 4649045868, 14721978582, 46490458680, 146444944842, 460255540932, 1443528742014, 4518872583696, 14121476824050, 44059007691036

Offset: 1

Stanislav Sykora, May 24 2012

Main transitions in systems of n particles with spin 1.
Consider the set S of all b^n numbers which have n digits in base b. Define as "main transition" a pair (x,y) of elements of S such that x and y differ in base b in only one digit which in y exceeds that in x by 1. This particular sequence a(n) gives the number of such transitions for the case b=3.
The terminology originates from quantum theory of coupled spin systems (such as in magnetic resonance) with n particles, each with spin S = (b-1)/2. Then the i-th digit's value in base b can be intended as a label for the b = 2S+1 quantum states of the i-th particle. The most intense main quantum transitions then correspond to the above definition. Due to continuity, the correspondence holds regardless of how strongly coupled are the particles among themselves.
a(n) is the number of functions from {1,2,...,n} into {1,2,3} with a specially designated element of the domain that is restricted to be mapped into {1,2}. Hence the e.g.f. is 2*x*exp(x)^3. - Geoffrey Critzer, Mar 01 2015
a(n) is the distance spectral radius of the dimension-regular generalized recursive circulant graph (commonly known as multiplicative circulant graph) of order 3^n. - John Rafael M. Antalan, Sep 25 2020

			n=2, b=3, S={00, 01, 02, 10, 11, 12, 20, 21, 22}, main transitions = {(00,01), (00,10), (01,02), (01,12), (02,12), (10,11), (10,20), (11,12), (11,21), (12,22), (20,21), (21,22)}, main transitions count = 12.

M. H. Levitt, Spin Dynamics, Basics of Nuclear Magnetic Resonance, 2nd Edition, John Wiley & Sons, 2007, Section 6 (Mathematical techniques).
J. A. Pople, W. G. Schneider, H. J. Bernstein, High-Resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.

Stanislav Sykora, Table of n, a(n) for n = 1..100
John Rafael M. Antalan and Francis Joseph H. Campeña, Distance eigenvalues and forwarding indices of dimension-regular generalized recursive circulant graph of order power of two and three, arXiv:2009.11608[math.CO], 2020.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A001787 (b = 2).
Cf. A212698, A212699, A212700, A212701, A212702, A212703, A212704 (b = 4, 5, 6, 7, 8, 9, 10).
Row n=3 of A258997.

List([1..30], n-> 2*3^(n-1)*n) # G. C. Greubel, Jun 08 2019

[2*3^(n-1)*n: n in [1..30]]; // G. C. Greubel, Jun 08 2019

A212697:=n->2*n*3^(n-1): seq(A212697(n), n=1..30); # Wesley Ivan Hurt, Mar 01 2015

Table[Sum[Binomial[n, j] j 2^j, {j, n}], {n, 30}] (* Geoffrey Critzer, Mar 01 2015 *)
Table[2*3^(n-1)*n, {n,30}] (* G. C. Greubel, Jun 08 2019 *)

mtrans(n,b) = n*(b-1)*b^(n-1);
for (n=1,100,write("b212697.txt",n," ",mtrans(n,3)))

[2*3^(n-1)*n for n in (1..30)] # G. C. Greubel, Jun 08 2019

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=3.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 2*x/(1-3*x)^2.
a(n) = 2*A027471(n+1). (End)
a(n) = A005843(n)*A000244(n-1). - Omar E. Pol, Jan 21 2014
a(n) = Sum_{j=1..n} binomial(n,j)*j*2^j. - Geoffrey Critzer, Mar 01 2015
E.g.f.: 2*x*exp(3*x). - G. C. Greubel, Jun 08 2019

A212700 a(n) = 5n6^(n-1).

Original entry on oeis.org

5, 60, 540, 4320, 32400, 233280, 1632960, 11197440, 75582720, 503884800, 3325639680, 21767823360, 141490851840, 914248581120, 5877312307200, 37614798766080, 239794342133760, 1523399350026240, 9648195883499520, 60935974001049600, 383896636206612480, 2413064570441564160

Offset: 1

Stanislav Sykora, May 25 2012

Main transitions in systems of n particles with spin 5/2.
Refer to the general explanation in A212697.
This particular sequence is obtained for base b=6, corresponding to spin S=(b-1)/2=5/2.
Arithmetic derivative of 6^n: a(n) = A003415(6^n). - Bruno Berselli, Oct 22 2013

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A001787, A212697, A212698, A212699, A212701, A212702, A212703, A212704 (b = 2, 3, 4, 5, 7, 8, 9, 10).

Cf. A000400, A003415, A008587, A053469.

Rest@ CoefficientList[Series[5 x/(6 x - 1)^2, {x, 0, 18}], x] (* or *)
Array[5 # 6^(# - 1) &, 18] (* Michael De Vlieger, Nov 18 2019 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212700.txt", n, " ", mtrans(n, 6)))

a(n) = n*(b-1)*b^(n-1): for this sequence, set b=6.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 5*x/(6*x-1)^2.
a(n) = 5*A053469(n). (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 5*x*exp(6*x).
a(n) = A008587(n)*A000400(n-1).
a(n) = 12*a(n-1) - 36*a(n-2) for n > 2. (End)

A002699 a(n) = n2^(2n-1).

Original entry on oeis.org

0, 2, 16, 96, 512, 2560, 12288, 57344, 262144, 1179648, 5242880, 23068672, 100663296, 436207616, 1879048192, 8053063680, 34359738368, 146028888064, 618475290624, 2611340115968, 10995116277760, 46179488366592, 193514046488576

Offset: 0

N. J. A. Sloane

Right side of binomial sum Sum(i * binomial(2*n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
Coefficients of shifted Chebyshev polynomials.
Starting with offset 1 = 4th binomial transform of [2, 8, 0, 0, 0, ...]. - Gary W. Adamson, Jul 21 2009
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 symmetric difference of x and y for every (x,y) of B. - Ross La Haye, Jan 04 2013
It's the relation [27] with T(n) in the document of Ross. Following the last comment of Ross, A002697 is the similar sequence when replacing "symmetric difference" by "intersection" and A212698 is the similar  sequence when replacing "symmetric difference" by union. - Bernard Schott, Jan 04 2013
If Delta = Symmetric difference, here, X Delta Y and Y Delta X are considered as two distinct Cartesian products, if we want to consider that X Delta Y = X Delta Y is the same Cartesian product, see A002697. - Bernard Schott, Jan 15 2013

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
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 = 0..1000
Rebecca Bourn and William Q. Erickson, A palindromic polynomial connecting the earth mover's distance to minuscule lattices of Type A, arXiv:2307.02652 [math.CO], 2023.
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)
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
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Found in A053125 and A053124.

Cf. A000051, A002457, A002697, A212698.

[n*2^(2*n-1): n in [0..30]]; /* or */ I:=[0, 2]; [n le 2 select I[n] else 8*Self(n-1)-16*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2013

A002699 := n->n*2^(2*n-1);
A002699:=2*z/(4*z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

Table[(n 2^(2 n - 1)), {n, 0, 30}] (* Vincenzo Librandi, Mar 20 2013 *)
LinearRecurrence[{8,-16},{0,2},30] (* Harvey P. Dale, Dec 20 2015 *)

a(n)=n*2^(2*n-1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2 * A002697(n). - Bernard Schott, Jan 04 2013
a(n) = A212698(n) - A002697(n)
a(n) = 8*a(n-1)-16*a(n-2) with n>1, a(0)=0, a(1)=2. - Vincenzo Librandi, Mar 20 2013
G.f.: (2*x)/(1 - 4*x)^2. - Harvey P. Dale, Jul 28 2021
E.g.f.: (exp(4*x) - 1)/2. - Stefano Spezia, Aug 04 2022

A212703 Main transitions in systems of n particles with spin 4.

Original entry on oeis.org

8, 144, 1944, 23328, 262440, 2834352, 29760696, 306110016, 3099363912, 30993639120, 306837027288, 3012581722464, 29372671794024, 284688972772848, 2745215094595320, 26354064908115072, 252010745683850376, 2401514164751985936, 22814384565143866392, 216136274827678734240

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=9 (see formula), corresponding to spin S=(b-1)/2=4.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212702, A212704 (b = 2, 3, 4, 5, 6, 7, 8, 10).

Cf. A001019, A008590, A053540.

LinearRecurrence[{18,-81},{8,144},30] (* Harvey P. Dale, Jun 28 2017 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212703.txt", n, " ", mtrans(n, 9)))

Vec(8*x/(9*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n)=8*n*9^(n-1) \\ Charles R Greathouse IV, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=9.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) for n > 2.
G.f.: 8*x/(9*x-1)^2. (End)
From Elmo R. Oliveira, May 13 2025: (Start)
E.g.f.: 8*x*exp(9*x).
a(n) = 8*A053540(n) = A008590(n)*A001019(n-1). (End)

A212699 Main transitions in systems of n particles with spin 2.

Original entry on oeis.org

4, 40, 300, 2000, 12500, 75000, 437500, 2500000, 14062500, 78125000, 429687500, 2343750000, 12695312500, 68359375000, 366210937500, 1953125000000, 10375976562500, 54931640625000, 289916992187500, 1525878906250000, 8010864257812500, 41961669921875000, 219345092773437500

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This particular sequence is obtained for base b=5, corresponding to spin S=(b-1)/2=2.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A001787, A212697, A212698, A212700, A212701, A212702, A212703, A212704 (b = 2, 3, 4, 6, 7, 8, 9, 10).

Cf. A000351, A008586, A053464.

Join[{4},Table[4n*5^(n-1),{n,20}]] (* or *) Join[{4},LinearRecurrence[{10,-25},{4,40},20]] (* Harvey P. Dale, Aug 19 2014 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212699.txt", n, " ", mtrans(n, 5)))

a(n) = n*(b-1)*b^(n-1) where b=5.
a(n) = 10*a(n-1) - 25*a(n-2), a(0)=a(1)=4, a(2)=40. - Harvey P. Dale, Aug 19 2014
From Elmo R. Oliveira, May 13 2025: (Start)
G.f.: 4*x/(5*x-1)^2.
E.g.f.: 4*x*exp(5*x).
a(n) = 4*A053464(n) = A008586(n)*A000351(n-1). (End)

A212701 Main transitions in systems of n particles with spin 3.

Original entry on oeis.org

6, 84, 882, 8232, 72030, 605052, 4941258, 39530064, 311299254, 2421216420, 18643366434, 142367525496, 1079620401678, 8138676874188, 61040076556410, 455765904954528, 3389758918099302, 25124095510618356, 185639150161791186, 1367867422244777160, 10053825553499112126

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=7 (see formula), corresponding to spin S=(b-1)/2=3.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A001787, A212697, A212698, A212699, A212700, A212702, A212703, A212704 (b = 2, 3, 4, 5, 6, 8, 9, 10).

Cf. A000420, A008588, A027473.

LinearRecurrence[{14,-49},{6,84},20] (* Harvey P. Dale, Aug 02 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212701.txt", n, " ", mtrans(n, 7)))

Vec(6*x/(7*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=7.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 14*a(n-1) - 49*a(n-2) for n > 2.
G.f.: 6*x/(7*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 6*x*exp(7*x).
a(n) = 6*A027473(n) = A008588(n)*A000420(n-1). (End)

A212702 Main transitions in systems of n particles with spin 7/2.

Original entry on oeis.org

7, 112, 1344, 14336, 143360, 1376256, 12845056, 117440512, 1056964608, 9395240960, 82678120448, 721554505728, 6253472382976, 53876069761024, 461794883665920, 3940649673949184, 33495522228568064, 283726776524341248, 2395915001761103872, 20176126330619822080

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=8 (see formula), corresponding to spin S=(b-1)/2=7/2.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212703, A212704 (b = 2, 3, 4, 5, 6, 7, 9, 10).

Cf. A001018, A008589, A053539.

LinearRecurrence[{16,-64},{7,112},30] (* Harvey P. Dale, Feb 11 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212702.txt", n, " ", mtrans(n, 8)))

Vec(7*x/(8*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=8.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 16*a(n-1) - 64*a(n-2) for n > 2.
G.f.: 7*x/(8*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 7*x*exp(8*x).
a(n) = 7*A053539(n) = A008589(n)*A001018(n-1). (End)

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

cp's OEIS Frontend

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

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A212704 a(n) = 9*n*10^(n-1).

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A212697 a(n) = 2*n*3^(n-1).

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A212700 a(n) = 5*n*6^(n-1).

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A002699 a(n) = n*2^(2*n-1).

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A212703 Main transitions in systems of n particles with spin 4.

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A212704 a(n) = 9n10^(n-1).

A212697 a(n) = 2n3^(n-1).

A212700 a(n) = 5n6^(n-1).

A002699 a(n) = n2^(2n-1).