A151628 Number of permutations of 2 indistinguishable copies of 1..n with exactly 6 adjacent element pairs in decreasing order.

0, 0, 0, 1, 5158, 1822014, 242384856, 19323413187, 1130781824398, 54076536713976, 2251621794635088, 84973986733001061, 2985450779006443846, 99474230412387811666, 3185003930126491696920, 98939258210106714816135, 3003063241991742340646382, 89537653738976723063722828

Offset

1,5

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (210, -20559, 1249006, -52877484, 1660698792, -40217937324, 770684131800, -11898983656350, 149952459677980, -1556983686224082, 13409725967210820, -96243862494272068, 577309836510214632, -2898323263572570108, 12179135493109203192, -42783931230910840233, 125321632824303394722, -304873695791112375063, 612528671944484732862, -1008704075532213688776, 1347947980011397405152, -1442243815707288575520, 1213235943921688622400, -782048981957040864000, 371788922872056960000, -122537228378997600000, 24957111340728000000, -2362404048480000000).

Formula

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

a(n) = Sum_{j=0..6} (-1)^j*binomial(2*n+1, j)*binomial(8-j, 2)^n.

G.f.: x^4*(1 + 4948*x + 759393*x^2 - 35443768*x^3 - 508116211*x^4 + 51430255228*x^5 - 1039884450243*x^6 + 5791934217096*x^7 + 99233948186819*x^8 - 2137209451932636*x^9 + 17699047175646675*x^10 - 64844223652304424*x^11 - 67279992193011969*x^12 + 1850800989665593044*x^13 - 8839633922267140593*x^14 + 20366483030687973816*x^15 - 15348635039953199376*x^16 - 39686222209918929480*x^17 + 123668352881463084480*x^18 - 135232901326862200800*x^19 + 35906630373023328000*x^20 + 48364304383014480000*x^21 - 29287301536936800000*x^22 - 4134207084840000000*x^23)/((1-x)^7*(1-3*x)^6*(1-6*x)^5*(1-10*x)^4*(1-15*x)^3*(1-21*x)^2*(1-28*x)).

E.g.f.: exp(28*x) - (1 + 42*x)*exp(21*x) + 45*x*(1 + 10*x)*exp(15*x) - (10/3)*x*(3 + 120*x + 400*x^2)*exp(10*x) + 18*x^2*(5 + 40*x + 48*x^2)*exp(6*x) - (9/10)*x^2*(5 + 90*x + 180*x^2 + 72*x^3)*exp(3*x) + (1/90)*x^3*(105 + 210*x + 84*x^2 + 8*x^3)*exp(x). (End)

Mathematica

With[{B=Binomial}, Table[Sum[(-1)^j*B[2n+1, j]*B[8-j, 2]^n, {j, 0, 6}], {n, 30}]] (* G. C. Greubel, Sep 07 2022 *)

Prog

(Magma) [(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(8-j, 2)^n: j in [0..6]]): n in [1..30]]; // G. C. Greubel, Sep 07 2022
(SageMath)
@CachedFunction
def A151628(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(8-j, 2)^n for j in (0..6))
[A151628(n) for n in (1..30)] # G. C. Greubel, Sep 07 2022

Crossrefs

Column k=6 of A154283.

Sequence in context: A213701 A224462 A251679 * A210400 A073585 A249841

Adjacent sequences: A151625 A151626 A151627 * A151629 A151630 A151631

Keyword

nonn

Author

R. H. Hardin, May 29 2009

Extensions

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

Status

approved