A384506 -id:A384506 - cp's OEIS frontend

A384686 a(n) = 2^(n-4)(5binomial(n,5) + 6*binomial(n,4)).

0, 0, 0, 0, 6, 70, 480, 2520, 11200, 44352, 161280, 549120, 1774080, 5491200, 16400384, 47523840, 134184960, 370442240, 1002700800, 2667184128, 6985482240, 18042716160, 46022000640, 116064256000, 289696382976, 716282265600, 1755735654400, 4269382041600, 10305404928000

Offset: 0

Enrique Navarrete, Jun 07 2025

a(n) is the number of words of length n defined on 5 letters that have exactly two a's and exactly two b's and no c's or exactly two a's and exactly three c's and no b's.

			a(4) = 6 since the words are the 6 permutations of aabb.
a(6) = 480 since the words are the 90 permutations of aabbdd, the 180 permutations of aabbde, the 90 permutations of aabbee, the 60 permutations of aacccd, and the 60 permutations of aaccce.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Cf. A384506.

A384686[n_] := 2^(n - 4)*(5*Binomial[n, 5] + 6*Binomial[n, 4]);
Array[A384686, 30, 0] (* Paolo Xausa, Jun 13 2025 *)
LinearRecurrence[{12,-60,160,-240,192,-64},{0,0,0,0,6,70},40] (* Harvey P. Dale, Jul 19 2025 *)

a(n) = 1/3*2^(n-7)*(n-3)*(n-2)*(n-1)*n*(n+2).
E.g.f.: x^2/2*exp(2*x)*(x^2/2 + x^3/6).
G.f.: 2*x^4*(3 - x)/(1 - 2*x)^6. - Stefano Spezia, Jun 07 2025

A384536 a(n) = 4^n - 2^(n-6)15binomial(n,6).

Original entry on oeis.org

1, 4, 16, 64, 256, 1024, 4081, 16174, 63856, 252064, 998176, 3972544, 15890176, 63814144, 256903936, 1035303424, 4171964416, 16799678464, 67578904576, 271543926784, 1089985970176, 4371374669824, 17518838480896, 70170274299904, 280945723703296

Offset: 0

Enrique Navarrete, Jun 02 2025

a(n) is the number of strings of length n defined on {0, 1, 2, 3} that do not contain exactly two 2's and exactly four 3's.

			a(8) = 63856 since from the 65536 strings of length 8 we subtract the 420 permutations of 33332200, the 840 permutations of 33332201 and the 420 permutations of 33332211.

Index entries for linear recurrences with constant coefficients, signature (18,-140,616,-1680,2912,-3136,1920,-512).

Cf. A384506.

a[n_]:=4^n-2^(n-6)*15*Binomial[n,6];Array[a,25,0] (* or *)
LinearRecurrence[{18,-140,616,-1680,2912,-3136,1920,-512},{1, 4, 16, 64, 256, 1024, 4081, 16174},25] (* or *)
CoefficientList[Series[ (1 - 14*x + 84*x^2 - 280*x^3 + 560*x^4 - 672*x^5 + 433*x^6 - 68*x^7)/((1 - 2*x)^7*(1 - 4*x)),{x,0,24}],x] (* James C. McMahon, Jun 08 2025 *)

G.f.: (1 - 14*x + 84*x^2 - 280*x^3 + 560*x^4 - 672*x^5 + 433*x^6 - 68*x^7)/((1 - 2*x)^7*(1 - 4*x)). - Stefano Spezia, Jun 02 2025

cp's OEIS Frontend

A384686 a(n) = 2^(n-4)(5binomial(n,5) + 6*binomial(n,4)).

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A384536 a(n) = 4^n - 2^(n-6)15binomial(n,6).

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cp's OEIS Frontend

A384686 a(n) = 2^(n-4)*(5*binomial(n,5) + 6*binomial(n,4)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A384536 a(n) = 4^n - 2^(n-6)*15*binomial(n,6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A384686 a(n) = 2^(n-4)(5binomial(n,5) + 6*binomial(n,4)).

A384536 a(n) = 4^n - 2^(n-6)15binomial(n,6).