A151624 Number of permutations of 2 indistinguishable copies of 1..n with exactly 2 adjacent element pairs in decreasing order.

0, 1, 48, 603, 5158, 37257, 247236, 1568215, 9703890, 59226357, 358722928, 2163496611, 13017647646, 78225458401, 469740168924, 2819689366191, 16922139539626, 101545622110989, 609314411814024, 3656015481903355, 21936500845191030, 131620291694585721

Offset: 1

Author

R. H. Hardin, May 29 2009

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (15,-84,226,-309,207,-54).

Crossrefs

Column k=2 of A154283.

Programs

Magma

[(&+[(-1)^j*Binomial(2*n+1,2-j)*Binomial(j+2,2)^n: j in [0..2]]): n in [1..40]]; // G. C. Greubel, Jun 19 2022

Mathematica

Table[6^n -(2*n+1)*3^n +n*(2*n+1), {n,40}] (* G. C. Greubel, Jun 19 2022 *)

PARI

a(n) = {6^n - (2*n + 1)*3^n + n*(2*n + 1)} \\ Andrew Howroyd, May 06 2020

PARI

Vec(x^2*(1 + 33*x - 33*x^2 - 81*x^3) / ((1 - x)^3*(1 - 3*x)^2*(1 - 6*x)) + O(x^25)) \\ Colin Barker, Jul 16 2020

SageMath

[6^n -(2*n+1)*3^n +binomial(2*n+1,2) for n in (1..40)] # G. C. Greubel, Jun 19 2022

Formula

a(n) = 6^n - (2*n + 1)*3^n + n*(2*n + 1). - Andrew Howroyd, May 06 2020

From Colin Barker, Jul 16 2020: (Start)

G.f.: x^2*(1 + 33*x - 33*x^2 - 81*x^3) / ((1 - x)^3*(1 - 3*x)^2*(1 - 6*x)).

a(n) = 15*a(n-1) - 84*a(n-2) + 226*a(n-3) - 309*a(n-4) + 207*a(n-5) - 54*a(n-6) for n>6.

(End)

E.g.f.: x*(3+2*x)*exp(x) - (1+6*x)*exp(3*x) + exp(6*x). - G. C. Greubel, Jun 19 2022

Extensions

Terms a(12) and beyond from Andrew Howroyd, May 06 2020