OFFSET
0,2
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.
LINKS
Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).
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
EXAMPLE
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.
MATHEMATICA
CoefficientList[Series[1/(1-4*x-6*x^2-4*x^3-x^4), {x, 0, 23}], x] (* Stefano Spezia, Jul 09 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jul 08 2024
STATUS
approved