A151652 Number of permutations of 6 indistinguishable copies of 1..n with exactly 3 adjacent element pairs in decreasing order.

0, 400, 233300, 35138736, 3656408776, 333546436832, 28929582376716, 2460253392551584, 207633993378572384, 17472055390759340592, 1468614660871788474628, 123393355120299525767120, 10365952122947333091123000, 870767651291136123846763456, 73145318723749749228550353212

Offset

1,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (165,-9225,237297,-3097935,21760767,-83084547, 168886683,-184022244,101917648,-22588608).

Formula

a(n) = 84^n - (6*n + 1)*28^n + binomial(6*n+1, 2)*7^n - binomial(6*n+1, 3). - Andrew Howroyd, May 07 2020

From G. C. Greubel, Sep 12 2022: (Start)

a(n) = Sum_{j=0..3} (-1)^j*binomial(6*n+1, j)*binomial(9-j, 6)^n.

G.f.: 4*x^2*(100 + 41825*x + 83559*x^2 - 21052241*x^3 + 67903073*x^4 + 129637536*x^5 - 671540492*x^6 - 210827008*x^7)/( Product_{j=0..3} (1 - binomial(j+6, 6)*x)^(4-j) ).

E.g.f.: exp(84*x) - (1 + 168*x)*exp(28*x) + 147*x*(1 + 6*x)*exp(7*x) - x*(35 + 108*x + 36*x^2)*exp(x). (End)

Mathematica

Table[Sum[(-1)^j*Binomial[6*n+1, j]*Binomial[9-j, 6]^n, {j, 0, 3}], {n, 30}] (* G. C. Greubel, Sep 12 2022 *)

Prog

(PARI) a(n) = {84^n - (6*n + 1)*28^n + binomial(6*n+1, 2)*7^n - binomial(6*n+1, 3)} \\ Andrew Howroyd, May 07 2020
(Magma) [(&+[(-1)^j*Binomial(6*n+1, j)*Binomial(9-j, 6)^n: j in [0..3]]): n in [1..30]]; // G. C. Greubel, Sep 12 2022
(SageMath)
def A151652(n): return sum((-1)^j*binomial(6*n+1, j)*binomial(9-j, 6)^n for j in (0..3))
[A151652(n) for n in (1..30)] # G. C. Greubel, Sep 12 2022

Crossrefs

Column k=3 of A237252.

Sequence in context: A210262 A151608 A022232 * A197441 A268366 A013767

Adjacent sequences: A151649 A151650 A151651 * A151653 A151654 A151655

Keyword

nonn

Author

R. H. Hardin, May 29 2009

Extensions

Terms a(7) and beyond from Andrew Howroyd, May 07 2020

Status

approved