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A374454
Expansion of o.g.f. 1/(1 - 4*x - 6*x^2 - 4*x^3 - x^4).
2
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
0,2
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.
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.
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