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%I #19 Jun 12 2023 07:38:25
%S 0,0,0,1,5158,1822014,242384856,19323413187,1130781824398,
%T 54076536713976,2251621794635088,84973986733001061,
%U 2985450779006443846,99474230412387811666,3185003930126491696920,98939258210106714816135,3003063241991742340646382,89537653738976723063722828
%N Number of permutations of 2 indistinguishable copies of 1..n with exactly 6 adjacent element pairs in decreasing order.
%H Andrew Howroyd, <a href="/A151628/b151628.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, 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).
%F From _G. C. Greubel_, Sep 07 2022: (Start)
%F a(n) = Sum_{j=0..6} (-1)^j*binomial(2*n+1, j)*binomial(8-j, 2)^n.
%F 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)).
%F 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)
%t 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 *)
%o (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
%o (SageMath)
%o @CachedFunction
%o def A151628(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(8-j, 2)^n for j in (0..6))
%o [A151628(n) for n in (1..30)] # _G. C. Greubel_, Sep 07 2022
%Y Column k=6 of A154283.
%K nonn
%O 1,5
%A _R. H. Hardin_, May 29 2009
%E Terms a(11) and beyond from _Andrew Howroyd_, May 06 2020
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