OFFSET
0,2
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.
LINKS
Index entries for linear recurrences with constant coefficients, signature (5,10,10,5,1).
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
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 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.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jul 08 2024
STATUS
approved