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

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

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 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

• Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).

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