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

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 (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).

Crossrefs

Column k=6 of A154283.

Programs

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

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 *)

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

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)

Extensions

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