A151647 Number of permutations of 5 indistinguishable copies of 1..n with exactly 2 adjacent element pairs in decreasing order.

0, 100, 5925, 167475, 3882250, 84320250, 1791011475, 37753995925, 793816473600, 16676797204500, 350257183908625, 7355694727665975, 154471515733316550, 3243914368665860350, 68122282848892857375, 1430568461732082827625, 30041941039388979651100

Offset

1,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (36,-390,1720,-3165,2556,-756).

Formula

a(n) = 21^n - (5*n + 1)*6^n + 5*n*(5*n + 1)/2. - Andrew Howroyd, May 06 2020

From Colin Barker, Jul 18 2020: (Start)

G.f.: 25*x^2*(4 + 93*x - 273*x^2 - 324*x^3)/((1 - x)^3*(1 - 6*x)^2*(1 - 21*x)).

a(n) = 36*a(n-1) - 390*a(n-2) + 1720*a(n-3) - 3165*a(n-4) + 2556*a(n-5) - 756*a(n-6) for n>6. (End)

From G. C. Greubel, Sep 12 2022: (Start)

a(n) = Sum_{j=0..2} (-1)^j*binomial(5*n+1, j)*binomial(7-j, 5)^n.

E.g.f.: exp(21*x) - (1 + 30*x)*exp(6*x) + (5/2)*x*(6 + 5*x)*exp(x). (End)

Mathematica

LinearRecurrence[{36, -390, 1720, -3165, 2556, -756}, {0, 100, 5925, 167475, 3882250, 84320250}, 30] (* Harvey P. Dale, Nov 01 2021 *)

Prog

(PARI) a(n) = {21^n -(5*n+1)*6^n +5*n*(5*n+1)/2} \\ Andrew Howroyd, May 06 2020
(PARI) concat(0, Vec(25*x^2*(4 + 93*x - 273*x^2 - 324*x^3) / ((1 - x)^3*(1 - 6*x)^2*(1 - 21*x)) + O(x^20))) \\ Colin Barker, Jul 18 2020
(Magma) [(&+[(-1)^j*Binomial(5*n+1, j)*Binomial(7-j, 5)^n: j in [0..2]]): n in [1..30]]; // G. C. Greubel, Sep 12 2022
(SageMath)
def A151647(n): return sum((-1)^j*binomial(5*n+1, j)*binomial(7-j, 5)^n for j in (0..2))
[A151647(n) for n in (1..30)] # G. C. Greubel, Sep 12 2022

Crossrefs

Column k=2 of A237202.

Sequence in context: A017763 A204081 A053109 * A245666 A210814 A065689

Adjacent sequences: A151644 A151645 A151646 * A151648 A151649 A151650

Keyword

nonn,easy

Author

R. H. Hardin, May 29 2009

Extensions

Terms a(8) and beyond from Andrew Howroyd, May 06 2020

Status

approved