A151641 Number of permutations of 4 indistinguishable copies of 1..n with exactly 3 adjacent element pairs in decreasing order.

0, 16, 8464, 724320, 37229920, 1558185200, 59416090096, 2167506244544, 77394535148480, 2734912695301840, 96159966699204560, 3372863224609356576, 118169571125488257824, 4137881135327148408240, 144857367811462402307760, 5070515828676757812456320

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (84,-2651,41784,-364146,1838880,-5445950,9357000,-9128125,4687500,-984375).

Formula

a(n) = 35^n - (4*n + 1)*15^n + binomial(4*n+1, 2)*5^n - binomial(4*n+1, 3). - Andrew Howroyd, May 07 2020

From G. C. Greubel, Sep 08 2022: (Start)

a(n) = Sum_{j=0..3} (-1)^j*binomial(4*n+1, j)*binomial(7-j, 4)^n.

G.f.: 16*x^2*(1 +445*x +3485*x^2 -115215*x^3 +200675*x^4 +798375*x^5 -1890625*x^6 -703125*x^7)/( Product_{j=0..3} (1 - binomial(j+4,4)*x)^(4-j) ).

E.g.f.: exp(35*x) -(1+60*x)*exp(15*x) +50*x*(1+4*x)*exp(5*x) -(2/3)*x*(15 +48*x +16*x^2)*exp(x). (End)

Mathematica

With[{B=Binomial}, Table[Sum[(-1)^j*B[4*n+1, j]*B[7-j, 4]^n, {j, 0, 3}], {n, 30}]] (* G. C. Greubel, Sep 08 2022 *)

Prog

(PARI) a(n) = {35^n - (4*n + 1)*15^n + binomial(4*n+1, 2)*5^n - binomial(4*n+1, 3)} \\ Andrew Howroyd, May 07 2020
(Magma) [(&+[(-1)^j*Binomial(4*n+1, j)*Binomial(7-j, 4)^n: j in [0..3]]): n in [1..30]]; // G. C. Greubel, Sep 08 2022
(SageMath)
def A151641(n): return sum((-1)^j*binomial(4*n+1, j)*binomial(7-j, 4)^n for j in (0..3))
[A151641(n) for n in (1..30)] # G. C. Greubel, Sep 08 2022

Crossrefs

Column k=3 of A236463.

Sequence in context: A069986 A366361 A265177 * A265240 A221137 A006448

Adjacent sequences: A151638 A151639 A151640 * A151642 A151643 A151644

Keyword

nonn

Author

R. H. Hardin, May 29 2009

Extensions

Terms a(8) and beyond from Andrew Howroyd, May 07 2020

Status

approved