A151648 Number of permutations of 5 indistinguishable copies of 1..n with exactly 3 adjacent element pairs in decreasing order.

0, 100, 52800, 6021225, 447069750, 28203920250, 1662432014600, 95167951614675, 5379642996402350, 302454892260579500, 16965492408059468000, 950717312310731109725, 53255101624187593866550, 2982626301173304400020350, 167034793385579317725373000, 9354122482050520106734846375

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 (120,-5135,105130,-1138875,6873212,-23367945,44115210,-45855180,24607800,-5334336).

Crossrefs

Column k=3 of A237202.

Programs

Magma

[(&+[(-1)^j*Binomial(5*n+1, j)*Binomial(8-j, 5)^n: j in [0..3]]): n in [1..30]]; // G. C. Greubel, Sep 12 2022

Mathematica

Table[Sum[(-1)^j*Binomial[5*n+1, j]*Binomial[8-j, 5]^n, {j,0,3}], {n, 30}] (* G. C. Greubel, Sep 12 2022 *)

PARI

a(n) = {56^n - (5*n + 1)*21^n + binomial(5*n+1, 2)*6^n - binomial(5*n+1, 3)} \\ Andrew Howroyd, May 07 2020

SageMath

def A151648(n): return sum((-1)^j*binomial(5*n+1, j)*binomial(8-j, 5)^n for j in (0..3))
[A151648(n) for n in (1..30)] # G. C. Greubel, Sep 12 2022

Formula

a(n) = 56^n - (5*n + 1)*21^n + binomial(5*n+1, 2)*6^n - binomial(5*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(5*n+1, j)*binomial(8-j, 5)^n.

G.f.: 25*x^2*(4 + 1632*x + 7949*x^2 - 594490*x^3 + 1502565*x^4 + 3945816*x^5 - 13945932*x^6 - 4667544*x^7)/( Product_{j=0.3} (1 - binomial(j+5, 5)*x)^(4-j) ).

E.g.f.: exp(56*x) - (1 + 105*x)*exp(21*x) + 90*x*(1 + 5*x)*exp(6*x) - (5/6)*x*(24 + 75*x + 25*x^2)*exp(x). (End)

Extensions

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