A088305 -id:A088305 - cp's OEIS frontend

A052623 E.g.f. x(1-x)^2/(1-3x+x^2).

0, 1, 2, 18, 192, 2520, 39600, 725760, 15200640, 358162560, 9376819200, 270037152000, 8483597337600, 288734500454400, 10582834303641600, 415593298568448000, 17408598098411520000, 774797125808369664000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 569

spec := [S,{S=Prod(Z,Sequence(Prod(Z,Sequence(Z),Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[x (1-x)^2/(1-3x+x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 30 2021 *)

E.g.f.: x*(-1+x)^2/(1-3*x+x^2)
Recurrence: {a(1)=1, a(0)=0, a(2)=2, (n^2+3*n+2)*a(n)+(-6-3*n)*a(n+1)+a(n+2)=0, a(3)=18}
Sum(-1/5*(-3+7*_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^2-3*_Z+1))*n!
a(n)=n!*A088305(n-1). - R. J. Mathar, Jun 03 2022

A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 3, 8, 27, 67, 216, 569, 1747, 4812, 14041, 39483, 115408, 326385, 941735, 2684170, 7725097, 22063737, 63354066, 181223899, 519883185, 1488316952, 4266788191, 12219763777, 35023995792, 100326757107, 287503501905, 823654031283, 2360146144917, 6761847714698, 19374935267810

Offset: 0

Ilya Gutkovskiy, May 24 2018

nonn

Invert transform of A163767.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A001906, A055887, A088305, A129373, A129374, A129921, A163767, A304963, A304964, A304965.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j$2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 24 2018

nmax = 30; CoefficientList[Series[1/(1 - Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).

A308446 Expansion of Product_{k>=1} 1/(1 - x^k)^Fibonacci(2*k).

Original entry on oeis.org

1, 1, 4, 12, 39, 118, 371, 1129, 3468, 10524, 31910, 96155, 289016, 865000, 2581577, 7679762, 22784896, 67418329, 199004329, 586052299, 1722165404, 5050349249, 14781877481, 43185726143, 125949155473, 366716549379, 1066057177765, 3094398005409, 8969054893842

Offset: 0

Ilya Gutkovskiy, May 27 2019

nonn

Euler transform of A001906.

Cf. A000045, A001906, A032170, A034691, A088305, A166861.

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^Fibonacci[2 k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Fibonacci[2 d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

a(n) ~ phi^(2*n) * exp(2*sqrt(n)/5^(1/4) - 3/10 + S) / (2 * 5^(1/8) * sqrt(Pi) * n^(3/4)), where S = Sum_{k>=2} 1/((phi^(2*k) - 3 + 1/phi^(2*k))*k) = 0.155349347463140787939176213528043741704916536093946010733676987281... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 28 2019

A308817 Total number of parts in all m-color dihedral compositions of n (that is, the total number of parts in all dihedral compositions of n where a part of size m may be colored with one of m colors).

Original entry on oeis.org

1, 4, 10, 26, 56, 138, 299, 726, 1686, 4158, 10130, 25678, 64725, 166538, 428456, 1112226, 2888604, 7533750, 19653903, 51367462, 134277878, 351284164, 919080550, 2405427698, 6295780309, 16480373968, 43141303978, 112939105716, 295664584064, 774042041090, 2026429360115, 5305210333758

Offset: 1

Petros Hadjicostas, Jun 26 2019

nonn

Cyclic compositions of a positive integer n are equivalence classes of ordered partitions of n such that two such partitions are equivalent iff one can be obtained from the other by rotation. These were first studied by Sommerville (1909).
Symmetric cyclic compositions or circular palindromes or achiral cyclic compositions are those cyclic compositions that have at least one axis of symmetry. They were also studied by Sommerville (1909, pp. 301-304).
Dihedral compositions of a positive integer n are equivalence classes of ordered partitions of n such that two such partitions are equivalent iff one can be obtained from the other by rotation or reflection. See, for example, Knopfmacher and Robbins (2013).
The number of dihedral compositions of n (of any kind) is the average of the corresponding numbers of cyclic and symmetric cyclic compositions of n.
Let c = (c(m): m >= 1) be the input sequence and let b_k = (b_k(n): n >= 1) be the output sequence under Bower's DIK[k] ("bracelet, indistinct, unlabeled" with k boxes) transform of c; that is, b_k(n) = (DIK[k] c)n for n >= 1. Hence, b_k(n) is the number of dihedral compositions of n with k parts, where a part of size m can be colored with one of c(m) colors. If C(x) = Sum{m >= 1} c(m)*x^m is the g.f. of the input sequence c, then the bivariate g.f. of the list of sequences (b_k: k >= 1) = ((DIK[k] c): k >= 1) = ((b_k(n): n >= 1): k >= 1) is Sum_{n,k >= 1} b_k(n)*x^n*y^k = (1 + y * C(x))^2/(4 * (1 - y^2 * C(x^2))) - (1/4) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - y^d * C(x^d)).
Here, Sum_{k=1..n} k*b_k(n) is the total number of parts in all colored dihedral compositions of n (where the coloring of parts is done according to the input sequence c). To find the g.f. of the sequence (Sum_{k=1..n} k*b_k(n): n >= 1), we differentiate the above bivariate g.f. (i.e., Sum_{n,k >= 1} b_k(n)*x^n*y^k) w.r.t. y and set y = 1. We get (1 + C(x)) * (C(x) + C(x^2))/(2 * (1 - C(x^2))^2) + (1/2) * Sum_{d >= 1} phi(d) * C(x^d)/(1 - C(x^d)).
For the current sequence, the input sequence is c(m) = m for m >= 1, and we are dealing with the so-called "m-color" compositions. m-color linear compositions were studied by Agarwal (2000), whereas m-color cyclic compositions were studied by Gibson (2017) and Gibson et al. (2018).
Thus, for the current sequence, a(n) is the total number of parts in all m-color dihedral compositions of n (where a part of size m may be colored with one of m colors). Since C(x) = x/(1 - x)^2, we get that (1 + C(x)) * (C(x) + C(x^2))/(2 * (1 - C(x^2))^2) + (1/2) * Sum_{d >= 1} phi(d) * C(x^d)/(1 - C(x^d))  = (1/2) * (x^2 - x + 1) * x * (x^2 + 3*x + 1) * (x + 1)^2/((x^2 + x - 1)^2 * (x^2 - x - 1)^2) + (1/2) * Sum_{s >= 1} phi(s) * x^s/(1 - 3*x^s + x^(2*s)).

			We have a(1) = 1 because 1_1 is the only m-color dihedral composition of n = 1 and the total number of parts is 1.
We have a(2) = 4 because 2_1, 2_2, 1_1 + 1_1 are all the m-color dihedral compositions of 2 and the total number of parts is 1 + 1 + 2 = 4.
We have a(3) = 10 because 3_1, 3_2, 3_3, 1_1 + 2_1, 1_1 + 2_2, 1_1 + 1_1 + 1_1 are all the m-color dihedral compositions of n = 3 and the total number of parts is 1 + 1 + 1 + 2 + 2 + 3 = 10.
We have a(4) = 26 because 4_1, 4_2, 4_3, 4_4, 1_1 + 3_1, 1_1 + 3_2, 1_1 + 3_3, 2_1 + 2_1, 2_1 + 2_2, 2_2 + 2_2, 1_1 + 2_1 + 1_1, 1_1 + 2_2 + 1_1, 1_1 + 1_1 + 1_1 + 1_1 are all the m-color dihedral compositions of n = 4 and the total number of parts is 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 4 = 26.
Note that a(n) = A307415(n) = A308723(n) for n = 1, 2, 3, 4 because all cyclic compositions of n when 1 <= n <= 4 are symmetric as well (and thus are dihedral).
For n = 5, we have A307415(5) = 53 <> 59 = A308723(5), in which case, a(n) = (53 + 59)/2 = 56. For example, 1_1 + 2_1 + 3_1 is a dihedral composition of n = 5, but it is not symmetric, so it corresponds to two (inequivalent) cyclic compositions: 1_1 + 2_1 + 3_1 and 3_1 + 2_1 + 1_1. Similarly, 1_1 + 2_1 + 2_2 is a dihedral composition of n = 5, but it is not symmetric, so it corresponds to two (inequivalent) cyclic compositions: 1_1 + 2_1 + 2_2 and 2_2 + 2_1 + 1_1.

A. K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.
C. G. Bower, Transforms (2).
Petros Hadjicostas, Generalized colored circular palindromic compositions, Moscow Journal of Combinatorics and Number Theory, 9(2) (2020), 173-186.
Meghann Moriah Gibson, Combinatorics of compositions, Master of Science, Georgia Southern University, 2017.
Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of n-color compositions, Discrete Mathematics 341 (2018), 3209-3226.
Arnold Knopfmacher and Neville Robbins, Some properties of dihedral compositions, Util. Math. 92 (2013), 207-220.
D. M. Y. Sommerville, On certain periodic properties of cyclic compositions of numbers, Proc. London Math. Soc. S2-7(1) (1909), 263-313.

Cf. A000045, A030267, A032198, A032287, A088305, A307415, A308723.

a(n) = (A307415(n) + A308723(n))/2 for n >= 1.
a(n) = (n/10) * (11*F(n) + 7*F(n-1)) + (-1)^n * (n/10) * (-4*F(n) + 7*F(n-1)) - (1/5) * (5*F(n+1) + 3*F(n)) - (-1)^n * (1/5) * (3*F(n) - 5*F(n-1)) + (1/2) * Sum_{s|n} phi(n/s) * F(2*s) for n >= 1, where F(n) = A000045(n) (Fibonacci numbers).
G.f.: (1/2) * (x^2 - x + 1) * x * (x^2 + 3*x + 1) * (x + 1)^2/((x^2 + x - 1)^2 * (x^2 - x - 1)^2) + (1/2) * Sum_{s >= 1} phi(s) * x^s/(1 - 3*x^s + x^(2*s)).

A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} ijx^(i*j)).

Original entry on oeis.org

1, 1, 5, 15, 53, 165, 561, 1807, 5993, 19586, 64491, 211466, 695101, 2281614, 7494995, 24610588, 80829373, 265437828, 871738976, 2862815763, 9401768055, 30875971366, 101399191222, 333001988025, 1093603789613, 3591473940515, 11794667169894, 38734550365835, 127207121681103, 417757532953031

Offset: 0

Ilya Gutkovskiy, Aug 27 2018

nonn

Cf. A000005, A038040, A088305, A129921, A180305, A280540, A280541.

a:=series(1/(1-add(add(i*j*x^(i*j),j=1..100),i=1..100)),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Apr 02 2019

nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[i j x^(i j), {i, 1, nmax}], {j, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k x^k/(1 - x^k)^2, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k DivisorSigma[0, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[k DivisorSigma[0, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

G.f.: 1/(1 - Sum_{k>=1} k*x^k/(1 - x^k)^2).
G.f.: 1/(1 - Sum_{k>=1} k*d(k)*x^k), where d(k) = number of divisors of k (A000005).
a(0) = 1; a(n) = Sum_{k=1..n} A038040(k)*a(n-k).
a(n) ~ c / r^n, where r = 0.304499876501217750838861744045680232405337905509126... is the root of the equation Sum_{k>=1} k*r^k/(1 - r^k)^2 = 1 and c = 0.44152042515136849968144466258954953693306684400261343177792428746297872748... - Vaclav Kotesovec, Aug 28 2018

A375100 Triangle read by rows: T(n,k) is the number of n-color compositions of n with k pairs of adjacent parts that are the same color.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 11, 6, 3, 1, 24, 18, 8, 4, 1, 53, 47, 26, 12, 5, 1, 118, 118, 79, 38, 17, 6, 1, 261, 297, 220, 122, 56, 23, 7, 1, 577, 740, 593, 370, 185, 80, 30, 8, 1, 1276, 1816, 1583, 1068, 589, 274, 111, 38, 9, 1, 2823, 4408, 4166, 3008, 1795, 908, 395, 150, 47, 10, 1

Offset: 1

John Tyler Rascoe, Jul 29 2024

			Triangle begins:
      k=0    1    2    3    4   5   6   7   8
 n=1:   1;
 n=2:   2,   1;
 n=3:   5,   2,   1;
 n=4:  11,   6,   3,   1;
 n=5:  24,  18,   8,   4,   1;
 n=6:  53,  47,  26,  12,   5,  1;
 n=7: 118, 118,  79,  38,  17,  6,  1;
 n=8: 261, 297, 220, 122,  56, 23,  7,  1;
 n=9: 577, 740, 593, 370, 185, 80, 30,  8,  1;
 ...
Row n = 3 counts:
T(3,0) = 5: (1,2_2), (2_2,1), (3_1), (3_2), (3_3).
T(3,1) = 2: (1,2_1), (2_1,1).
T(3,2) = 1: (1,1,1).

Cf. A088305 (row sums), A242551 (column k=0).

Cf. A003242, A372015, A374925.

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h= 1/(1-sum(i=1,N, x^i/(1-(x^i)*(y-1)-x)))); for(n=1, N-1, print(Vecrev(polcoeff(h, n))))}
T_xy(10)

G.f.: A(x,y) = 1/(1 - Sum_{i>0} (x^i)/(1 - (y-1)*x^i - x)).

cp's OEIS Frontend

A052623 E.g.f. x(1-x)^2/(1-3x+x^2).

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A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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A308446 Expansion of Product_{k>=1} 1/(1 - x^k)^Fibonacci(2*k).

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A308817 Total number of parts in all m-color dihedral compositions of n (that is, the total number of parts in all dihedral compositions of n where a part of size m may be colored with one of m colors).

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A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} i*j*x^(i*j)).

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A375100 Triangle read by rows: T(n,k) is the number of n-color compositions of n with k pairs of adjacent parts that are the same color.

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A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} ijx^(i*j)).