cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A108368 Coefficients of x/(1-3*x-3*x^2-x^3).

Original entry on oeis.org

0, 1, 3, 12, 46, 177, 681, 2620, 10080, 38781, 149203, 574032, 2208486, 8496757, 32689761, 125768040, 483870160, 1861604361, 7162191603, 27555258052, 106013953326, 407869825737, 1569206595241, 6037243216260, 23227219260240, 89362594024741, 343806683071203
Offset: 0

Views

Author

Michael Somos, Jun 01 2005

Keywords

Comments

From Enrique Navarrete, Jul 09 2024: (Start)
a(n+1) is the number of generalized compositions of n using parts of size at most 3 where there are binomial(3,i) types of i.
For example, the following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 5 using parts of size at most 3 where there are binomial(3,i) types of i (ie. 3 types of 1, 3 types of 2 and 1 type of 3):
Type Number Total
3+2 2 6
3+1+1 3 27
2+2+1 3 81
2+1+1+1 4 324
1+1+1+1+1 1 243,
adding to a(6) = 681.
The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i. (End).
For n>0, the expansion of (4^(1/3) + 2^(1/3) + 1)^n is a(n)*4^(1/3) + (a(n) + a(n-1))*2^(1/3) + (a(n) + 2*a(n-1) + a(n-2)). - Greg Dresden, Aug 14 2024

References

  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 562.

Links

Crossrefs

Cf. A000012, A000129, A374454, A374455.
Cf. A108369.

Programs

  • Mathematica
    CoefficientList[Series[x/(1-3*x-3*x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,3,1},{0,1,3},40] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)
  • PARI
    a(n)=if(n>=0, polcoeff(x/(1-3*x-3*x^2-x^3)+x*O(x^n),n), n=-1-n; polcoeff(x/(1+3*x+3*x^2-x^3)+x*O(x^n),n))

Formula

x=a(n), z=a(-n), y=a(n)+a(n-1), t=a(-n)+a(-n-1) is a solution to 2(x^3+z^3)=y^3+t^3.
G.f.: x/(1-3*x-3*x^2-x^3).
a(n) = 3*a(n-1)+3*a(n-2)+a(n-3).
a(-1-n) = A108369(n).
a(n+1) = Sum_{k>=0} (1/2)^(k+1) * binomial(3*k,n). - Seiichi Manyama, Aug 03 2024

A374454 Expansion of o.g.f. 1/(1 - 4*x - 6*x^2 - 4*x^3 - x^4).

Original entry on oeis.org

1, 4, 22, 116, 613, 3240, 17124, 90504, 478333, 2528092, 13361506, 70618412, 373233385, 1972618128, 10425707976, 55102092624, 291226324249, 1539193302772, 8134965235054, 42995028146468, 227237903531533, 1201000837247928, 6347545848001836, 33548135057767512
Offset: 0

Views

Author

Enrique Navarrete, Jul 08 2024

Keywords

Comments

a(n) is the number of generalized compositions of n using parts of size at most 4 where there are binomial(4,i) types of i (see example).
The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i.
Related sequences that count the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i are A108368(n+1), A000129(n+1), and A000012(n) for k = 3, 2, 1, respectively.

Examples

			The following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 6 using parts of size at most 4 where there are binomial(4,i) types of i (ie. 4 types of 1, 6 types of 2, 4 types of 3 and 1 type of 4):
    Type                     Number              Total
    4+2                        2                    12
    3+3                        1                    16
    4+1+1                      3                    48
    3+2+1                      6                   576
    2+2+2                      1                   216
    3+1+1+1                    4                  1024
    2+2+1+1                    6                  3456
    2+1+1+1+1                  5                  7680
    1+1+1+1+1+1                1                  4096,
    adding to a(6) = 17124.

Links

Crossrefs

Cf. A000012, A000129, A108368, A374455.
Cf. A145840.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-4*x-6*x^2-4*x^3-x^4),{x,0,23}],x] (* Stefano Spezia, Jul 09 2024 *)

Formula

a(n) = 4*a(n-1) + 6*a(n-2) + 4*a(n-3) + a(n-4), n=>4.
a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(4*k,n). - Seiichi Manyama, Aug 03 2024
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