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

Author

R. H. Hardin, May 29 2009

Keywords

• nonn

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).

Crossrefs

Column k=3 of A237252.

Programs

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

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 *)

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

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

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)

Extensions

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