A374455 Expansion of o.g.f. 1/(1 - 5*x - 10*x^2 - 10*x^3 - 5*x^4 - x^5).

1, 5, 35, 235, 1580, 10626, 71460, 480570, 3231845, 21734235, 146163251, 982951365, 6610371480, 44454906580, 298960311840, 2010515259876, 13520763292345, 90927457083265, 611489327404315, 4112280377388895, 27655184063541876, 185981775414350150, 1250731895575163300

Offset: 0

Author

Enrique Navarrete, Jul 08 2024

Keywords

• nonn
• easy

Comments

a(n) is the number of generalized compositions of n using parts of size at most 5 where there are binomial(5,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.

Examples

			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 5 where there are binomial(5,i) types of i (ie. 5 types of 1, 10 types of 2, 10 types of 3, 5 types of 4, and 1 type of 5):
    Type                     Number              Total
    5                          1                     1
    4+1                        2                    50
    3+2                        2                   200
    3+1+1                      3                   750
    2+2+1                      3                  1500
    2+1+1+1                    4                  5000
    1+1+1+1+1+1                1                  3125,
    adding to a(5)=10626.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5,10,10,5,1).

Crossrefs

Cf. A000012, A000129, A108368, A374454.

Cf. A145841.

Programs

Mathematica

CoefficientList[Series[1/(1-5*x-10*x^2-10*x^3-5*x^4-x^5),{x,0,22}],x] (* Stefano Spezia, Jul 09 2024 *)

Formula

a(n) = 5*a(n-1) + 10*a(n-2) + 10*a(n-3) + 5*a(n-4) + a(n-5), n=>5.

a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(5*k,n). - Seiichi Manyama, Aug 03 2024

Extensions

a(20) corrected by Georg Fischer, Oct 28 2024