A003520 -id:A003520 - cp's OEIS frontend

A276064 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,5} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/6)).

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 4, 2, 6, 3, 8, 3, 8, 4, 4, 12, 4, 4, 10, 12, 6, 16, 12, 5, 16, 24, 8, 24, 28, 6, 26, 40, 8, 10, 36, 52, 8, 8, 40, 60, 32, 13, 56, 84, 32, 11, 58, 96, 80, 17, 84, 136, 88, 15, 80, 160, 160, 16, 23, 120, 220, 192, 16

Offset: 0

Emeric Deutsch, Aug 22 2016

The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
Number of entries in row n is 1 + floor(n/6).
Sum of entries in row n is A003520(n).
T(n,0) = A226516(n+11).
Sum_{k>=0} k*T(n,k) = A276065(n).

			Row 8 is [1,4] because the compositions of 8 with parts in {1,5} are 5111, 1511, 1151, 1115 and 11111111, having asymmetry degrees 1, 1, 1, 1, and 0, respectively.
Triangle starts:
  1;
  1;
  1;
  1;
  1;
  2;
  1, 2;
  2, 2.

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A003520, A226516, A276065.

G := (1+z+z^5)/(1-z^2-2*t*z^6-z^10): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form

Table[TakeWhile[BinCounts[#, {0, 1 + Floor[n/4], 1}], # != 0 &] &@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; Nor[a == 1, a == 5]]], 1]]], {n, 0, 25}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)

G.f.: G(t,z) = (1+z+z^5)/(1-z^2-2*t*z^6-z^10). In the more general situation of compositions into a[1]=1} z^(a[j]), we have G(t,z) = (1 + F(z))/(1 - F(z^2) - t*(F(z)^2 - F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4k,3k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 57, 85, 122, 173, 249, 371, 575, 918, 1485, 2398, 3830, 6030, 9369, 14422, 22107, 33909, 52226, 80888, 125925, 196706, 307653, 480873, 750275, 1168085, 1815191, 2817518, 4371772, 6785606, 10539893, 16384908, 25488736

Offset: 0

Seiichi Manyama, Feb 28 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,1).

Cf. A003522, A100134, A137356.

Cf. A003520, A005689, A348289.

LinearRecurrence[{3, -3, 1, 0, 0, 0, 1}, Table[1, 7], 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\7, binomial(n-4*k, 3*k));

my(N=50, x='x+O('x^N)); Vec((1-x)^2/((1-x)^3-x^7))

G.f.: (1-x)^2/((1-x)^3 - x^7).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-7).

A052632 E.g.f. 1/(1-x-x^5).

Original entry on oeis.org

1, 1, 2, 6, 24, 240, 2160, 20160, 201600, 2177280, 29030400, 439084800, 7185024000, 124540416000, 2266635571200, 44460928512000, 941525544960000, 21341245685760000, 512189896458240000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 578

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

With[{nn=20},CoefficientList[Series[1/(1-x-x^5),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 13 2019 *)

E.g.f.: -1/(-1+x^5+x)
D-finite Recurrence: {a(1)=1, a(0)=1, a(3)=6, a(2)=2, a(4)=24, (-n^5-15*n^4-274*n-120-85*n^3-225*n^2)*a(n) +(-5-n)*a(n+4) +a(n+5)=0}
Sum(1/3381*(256+320*_alpha^4+400*_alpha^3+500*_alpha^2+625*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z^5+_Z))*n!
a(n)=n!*A003520(n). - R. J. Mathar, Jun 03 2022

A292027 a(n) = a(n-7) + a(n-11), starting a(0)=a(1)=...= a(10) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 7, 7, 7, 8, 9, 9, 9, 12, 12, 12, 13, 16, 16, 16, 20, 21, 21, 22, 28, 28, 28, 33, 37, 37, 38, 48, 49, 49, 55, 65, 65, 66, 81, 86, 86, 93, 113, 114, 115, 136, 151, 151, 159, 194, 200, 201, 229, 264, 265, 274

Offset: 0

Jason Bruce, Sep 07 2017

Kenneth H. Rosen, Discrete Mathematics and its Applications, McGraw-Hill, 2012, 501-503.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1,0,0,0,1).

Cf. A000930, A003269, A003520, A005708, A005709.

import java.util.Arrays;
public class IntegerSequences
{
    public static void main(String[] args)
    {
        int j = 7;
        int k = 11;
        // Set N to the number of terms you would like to generate.
        int N = 200;
        long[] G = new long[N];
        for(int i=0; i

LinearRecurrence[{0,0,0,0,0,0,1,0,0,0,1},{1,1,1,1,1,1,1,1,1,1,1},80] (* Harvey P. Dale, Oct 09 2018 *)

G.f.: (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/(1 - x^7 - x^11). - R. J. Mathar and N. J. A. Sloane, Nov 10 2017

A306489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{d|k} x^d).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 5, 1, 1, 1, 1, 3, 3, 8, 1, 1, 1, 2, 1, 6, 4, 13, 1, 1, 1, 1, 4, 1, 10, 6, 21, 1, 1, 1, 2, 1, 7, 2, 18, 9, 34, 1, 1, 1, 1, 3, 1, 13, 3, 31, 13, 55, 1, 1, 1, 2, 2, 6, 1, 25, 4, 55, 19, 89, 1, 1, 1, 1, 3, 3, 10, 1, 46, 5, 96, 28, 144, 1

Offset: 0

Ilya Gutkovskiy, Feb 19 2019

A(n,k) is the number of compositions (ordered partitions) of n into divisors of k.

			Square array begins:
  1,  1,  1,   1,  1,   1,  ...
  1,  1,  1,   1,  1,   1,  ...
  1,  2,  1,   2,  1,   2,  ...
  1,  3,  2,   3,  1,   4,  ...
  1,  5,  3,   6,  1,   7,  ...
  1,  8,  4,  10,  2,  13,  ...

Columns k=1..7 give A000012, A000045 (for n > 0), A000930, A060945, A003520, A079958, A005709.

Cf. A100346, A214575.

Table[Function[k, SeriesCoefficient[1/(1 - Sum[x^d, {d, Divisors[k]}]), {x, 0, n}]][i - n + 1], {i, 0, 12}, {n, 0, i}] // Flatten

G.f. of column k: 1/(1 - Sum_{d|k} x^d).

cp's OEIS Frontend

A276064 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,5} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/6)).

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A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4*k,3*k).

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PARI

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A052632 E.g.f. 1/(1-x-x^5).

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A292027 a(n) = a(n-7) + a(n-11), starting a(0)=a(1)=...= a(10) = 1.

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Java

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A306489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{d|k} x^d).

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Author

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Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4k,3k).