A151632 Number of permutations of 3 indistinguishable copies of 1..n with exactly 2 adjacent element pairs in decreasing order.

0, 9, 405, 6750, 83736, 922347, 9639783, 98361900, 992660346, 9967494609, 99857394225, 999379243674, 9997315646220, 99988457276295, 999950607877131, 9999789546603672, 99999106646803758, 999996220428781005, 9999984057081398901, 99999932929790707494

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (21,-153,503,-786,576,-160).

Formula

a(n) = 10^n - (3*n + 1)*4^n + 3*n*(3*n + 1)/2. - Andrew Howroyd, May 06 2020

From Colin Barker, Jul 17 2020: (Start)

G.f.: 9*x^2*(1 + 24*x - 42*x^2 - 64*x^3) / ((1 - x)^3*(1 - 4*x)^2*(1 - 10*x)).

a(n) = 21*a(n-1) - 153*a(n-2) + 503*a(n-3) - 786*a(n-4) + 576*a(n-5) - 160*a(n-6) for n>6.

(End)

Mathematica

T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*Binomial[3*n+1, k-j+2]*(Binomial[j+1, 3])^n, {j, 0, k+2}];
Table[T[n, 2], {n, 30}] (* G. C. Greubel, Mar 26 2022 *)

Prog

(PARI) a(n) = {10^n - (3*n + 1)*4^n + 3*n*(3*n + 1)/2} \\ Andrew Howroyd, May 06 2020
(PARI) concat(0, Vec(9*x^2*(1 + 24*x - 42*x^2 - 64*x^3) / ((1 - x)^3*(1 - 4*x)^2*(1 - 10*x)) + O(x^40))) \\ Colin Barker, Jul 17 2020
(Sage)
@CachedFunction
def T(n, k): return sum( (-1)^(k-j)*binomial(3*n+1, k-j+2)*(binomial(j+1, 3))^n for j in (0..k+2) )
[T(n, 2) for n in (1..30)] # G. C. Greubel, Mar 26 2022

Crossrefs

Column k=2 of A174266.

Sequence in context: A063876 A063156 A058850 * A035024 A179433 A091061

Adjacent sequences: A151629 A151630 A151631 * A151633 A151634 A151635

Keyword

nonn,easy

Author

R. H. Hardin, May 29 2009

Extensions

Terms a(10) and beyond from Andrew Howroyd, May 06 2020

Status

approved