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%I #12 Jun 11 2023 11:11:48
%S 0,0,8464,15018688,6501577152,1585757994496,290861341616496,
%T 45679059507623040,6563622028755987104,895009629522636673728,
%U 118277136569294999638992,15337783893522951844828992,1966328218272794506172178816,250347808250994150312231611520
%N Number of permutations of 4 indistinguishable copies of 1..n with exactly 5 adjacent element pairs in decreasing order.
%H Andrew Howroyd, <a href="/A151643/b151643.txt">Table of n, a(n) for n = 1..200</a>
%H G. C. Greubel, <a href="/A151643/a151643.txt">Generating functions and recurrence</a>
%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (462, -91531, 10359990, -751926160, 37211988696, -1302889937812, 33041329661136, -616216166660230, 8532037831808700, -88161849837783250, 681071135526667500, -3928678356172712500, 16849648606559250000, -53333569862180312500, 123260625734287500000, -205051485813712890625, 240815404573652343750, -193749765819873046875, 101292345769042968750, -30943180590820312500, 4187798876953125000).
%F From _G. C. Greubel_, Sep 10 2022: (Start)
%F a(n) = Sum_{j=0..5} (-1)^j*binomial(4*n+1, j)*binomial(9-j, 4)^n.
%F G.f., e.g.f., and recurrence are in the file "Generating functions and recurrence". (End)
%t Table[Sum[(-1)^j*Binomial[4*n+1, j]*Binomial[9-j, 4]^n, {j,0,5}], {n, 30}] (* _G. C. Greubel_, Sep 10 2022 *)
%o (Magma) [(&+[(-1)^j*Binomial(4*n+1, j)*Binomial(9-j, 4)^n: j in [0..5]]): n in [1..30]]; // _G. C. Greubel_, Sep 10 2022
%o (SageMath)
%o def A151643(n): return sum((-1)^j*binomial(4*n+1, j)*binomial(9-j, 4)^n for j in (0..5))
%o [A151643(n) for n in (1..30)] # _G. C. Greubel_, Sep 10 2022
%Y Column k=5 of A236463.
%K nonn
%O 1,3
%A _R. H. Hardin_, May 29 2009
%E Terms a(8) and beyond from _Andrew Howroyd_, May 06 2020
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