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

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 (21,-153,503,-786,576,-160).

Crossrefs

Column k=2 of A174266.

Programs

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 *)

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

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)

Extensions

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