A151626 Number of permutations of 2 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.

0, 0, 1, 603, 47290, 1822014, 49258935, 1086859301, 21147576440, 379269758400, 6441229796061, 105398150289775, 1680774708581766, 26324724399068626, 407112461900381715, 6239897666881158537, 95036195852840662820, 1440959515956284422196, 21778829725476446172249

Offset: 1

Author

R. H. Hardin, May 29 2009

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (70, -2163, 39122, -462914, 3792972, -22214806, 94629124, -295393077, 675442494, -1122134391, 1331169066, -1093400856, 588623760, -186332400, 26244000).

Crossrefs

Column k=4 of A154283.

Programs

Magma

A151626:= func< n | (&+[(-1)^(j+1)*Binomial(2*n+1,5-j)*Binomial(j+1,2)^n: j in [1..5]]) >;
[A151626(n): n in [1..30]]; // G. C. Greubel, Sep 07 2022

Mathematica

With[{B=Binomial}, Table[Sum[(-1)^(j+1)*B[2n+1, 5-j]*B[j+1,2]^n, {j,5}], {n,30}]] (* G. C. Greubel, Sep 07 2022 *)

PARI

a(n) = {15^n - (2*n + 1)*10^n + binomial(2*n+1, 2)*6^n - binomial(2*n+1, 3)*3^n + binomial(2*n+1, 4) } \\ Andrew Howroyd, May 07 2020

SageMath

@CachedFunction
def A151626(n): return sum((-1)^(j+1)*binomial(2*n+1,5-j)*binomial(j+1,2)^n for j in (1..5))
[A151626(n) for n in (1..30)] # G. C. Greubel, Sep 07 2022

Formula

a(n) = 15^n - (2*n + 1)*10^n + binomial(2*n+1, 2)*6^n - binomial(2*n+1, 3)*3^n + binomial(2*n+1, 4). - Andrew Howroyd, May 07 2020

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

G.f.: x^3*(1 + 533*x + 7243*x^2 - 223119*x^3 + 878573*x^4 + 5014923*x^5 - 40074183*x^6 + 75062403*x^7 + 19732086*x^8 - 185394420*x^9 + 117543960*x^10 + 43740000*x^11)/((1-x)^5*(1-3*x)^4*(1-6*x)^3*(1-10*x)^2*(1-15*x)).

E.g.f.: exp(15*x) - (1+20*x)*exp(10*x) + 18*x*(1+4*x)*exp(6*x) - 3*x*(1 + 12*x + 12*x^2)*exp(3*x) + (x^2/6)*(15 + 20*x + 4*x^2)*exp(x). (End)

Extensions

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