A151627 Number of permutations of 2 indistinguishable copies of 1..n with exactly 5 adjacent element pairs in decreasing order.

0, 0, 0, 72, 27664, 2864328, 163809288, 6727188848, 225167210712, 6590156148912, 175992170793456, 4407169187423736, 105396936343707456, 2437638848729751736, 55010494951127561400, 1219075824289276443744, 26652917330108137129544, 576864003740129587504224

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 (126, -7245, 252750, -5998905, 102950910, -1325335665, 13104159030, -101064788055, 614053303570, -2956203883287, 11303748373482, -34308033838515, 82354980144330, -155288585585115, 227560437228978, -255076375732668, 213622511296680, -128922602864400, 52813189404000, -13115963880000, 1488034800000).

Crossrefs

Column k=5 of A154283.

Programs

Magma

[(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(7-j, 2)^n: j in [0..5]]): n in [1..30]]; // G. C. Greubel, Sep 07 2022

Mathematica

With[{B = Binomial}, Table[Sum[(-1)^j*B[2n+1,j]*B[7-j,2]^n, {j,0,5}], {n, 30}]] (* G. C. Greubel, Sep 07 2022 *)

SageMath

@CachedFunction
def A151627(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(7-j, 2)^n for j in (0..5))
[A151627(n) for n in (1..30)] # G. C. Greubel, Sep 07 2022

Formula

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

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

G.f.: 8*x^4*(9 + 2324*x - 12462*x^2 - 1858545*x^3 + 34890010*x^4 - 134744022*x^5 - 1875623070*x^6 + 22965673068*x^7 - 95590873845*x^8 + 93562460910*x^9 + 576877450068*x^10 - 2203266593259*x^11 + 2865061552194*x^12 - 347005909980*x^13 - 2472141497400*x^14 + 1471264884000*x^15 + 318864600000*x^16)/((1-x)^6*(1-3*x)^5*(1-6*x)^4*(1-10*x)^3*(1-15*x)^2*(1-21*x)).

E.g.f.: exp(21*x) - (1 + 30*x)*exp(15*x) + 10*x*(3 + 20*x)*exp(10*x) - 6*x*(1 + 24*x + 48*x^2)*exp(6*x) + (9*x^2/2)*(5 + 20*x + 12*x^2)*exp(3*x) - (x^2/30)*(15 + 90*x + 60*x^2 + 8*x^3)*exp(x). (End)

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 06 2020