A151625 Number of permutations of 2 indistinguishable copies of 1..n with exactly 3 adjacent element pairs in decreasing order.

0, 0, 20, 1168, 27664, 450048, 6030140, 72338144, 811888600, 8742609264, 91700484132, 945739871600, 9647920866016, 97729381122976, 985430937715404, 9906926101717184, 99407643757824680, 996242539625306512, 9976235831375328564, 99850100363120616144

Offset

1,3

Links

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

Index entries for linear recurrences with constant coefficients, signature (35, -511, 4099, -19997, 61981, -123645, 157041, -122094, 52812, -9720).

Formula

a(n) = 10^n - (2*n + 1)*6^n + binomial(2*n+1, 2)*3^n - binomial(2*n+1, 3). - Andrew Howroyd, May 07 2020

From G. C. Greubel, Sep 07 2022: (Start)

G.f.: 4*x^3*(5 + 117*x - 749*x^2 - 831*x^3 + 6768*x^4 - 5022*x^5 - 3888*x^6)/((1-x)^4*(1-3*x)^3*(1-6*x)^2*(1-10*x)).

E.g.f.: exp(10*x) - (1+12*x)*exp(6*x) + 9*x*(1+2*x)*exp(3*x) - x*(3 + 12*x + 4*x^2)*exp(x)/3. (End)

Mathematica

With[{B=Binomial}, Table[Sum[(-1)^j*B[2n+1, 4-j]*B[j+1, 2]^n, {j, 4}], {n, 30}]] (* G. C. Greubel, Sep 07 2022 *)

Prog

(PARI) a(n) = {10^n - (2*n + 1)*6^n + binomial(2*n+1, 2)*3^n - binomial(2*n+1, 3)} \\ Andrew Howroyd, May 07 2020
(Magma)
A151625:= func< n | (&+[(-1)^j*Binomial(2*n+1, 4-j)*Binomial(j+1, 2)^n: j in [1..4]]) >;
[A151625(n): n in [1..30]]; // G. C. Greubel, Sep 07 2022
(SageMath)
@CachedFunction
def A151625(n): return sum((-1)^j*binomial(2*n+1, 4-j)*binomial(j+1, 2)^n for j in (1..4))
[A151625(n) for n in (1..30)] # G. C. Greubel, Sep 07 2022

Crossrefs

Column k=3 of A154283.

Sequence in context: A091535 A265654 A152130 * A203775 A184269 A042745

Adjacent sequences: A151622 A151623 A151624 * A151626 A151627 A151628

Keyword

nonn

Author

R. H. Hardin, May 29 2009

Extensions

Terms a(12) and beyond from Andrew Howroyd, May 07 2020

Status

approved