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%I #13 Sep 10 2022 07:33:06
%S 0,16,8464,724320,37229920,1558185200,59416090096,2167506244544,
%T 77394535148480,2734912695301840,96159966699204560,
%U 3372863224609356576,118169571125488257824,4137881135327148408240,144857367811462402307760,5070515828676757812456320
%N Number of permutations of 4 indistinguishable copies of 1..n with exactly 3 adjacent element pairs in decreasing order.
%H Andrew Howroyd, <a href="/A151641/b151641.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (84,-2651,41784,-364146,1838880,-5445950,9357000,-9128125,4687500,-984375).
%F a(n) = 35^n - (4*n + 1)*15^n + binomial(4*n+1, 2)*5^n - binomial(4*n+1, 3). - _Andrew Howroyd_, May 07 2020
%F From _G. C. Greubel_, Sep 08 2022: (Start)
%F a(n) = Sum_{j=0..3} (-1)^j*binomial(4*n+1, j)*binomial(7-j, 4)^n.
%F G.f.: 16*x^2*(1 +445*x +3485*x^2 -115215*x^3 +200675*x^4 +798375*x^5 -1890625*x^6 -703125*x^7)/( Product_{j=0..3} (1 - binomial(j+4,4)*x)^(4-j) ).
%F E.g.f.: exp(35*x) -(1+60*x)*exp(15*x) +50*x*(1+4*x)*exp(5*x) -(2/3)*x*(15 +48*x +16*x^2)*exp(x). (End)
%t With[{B=Binomial}, Table[Sum[(-1)^j*B[4*n+1,j]*B[7-j,4]^n, {j,0,3}], {n, 30}]] (* _G. C. Greubel_, Sep 08 2022 *)
%o (PARI) a(n) = {35^n - (4*n + 1)*15^n + binomial(4*n+1, 2)*5^n - binomial(4*n+1, 3)} \\ _Andrew Howroyd_, May 07 2020
%o (Magma) [(&+[(-1)^j*Binomial(4*n+1,j)*Binomial(7-j,4)^n: j in [0..3]]): n in [1..30]]; // _G. C. Greubel_, Sep 08 2022
%o (SageMath)
%o def A151641(n): return sum((-1)^j*binomial(4*n+1,j)*binomial(7-j,4)^n for j in (0..3))
%o [A151641(n) for n in (1..30)] # _G. C. Greubel_, Sep 08 2022
%Y Column k=3 of A236463.
%K nonn
%O 1,2
%A _R. H. Hardin_, May 29 2009
%E Terms a(8) and beyond from _Andrew Howroyd_, May 07 2020
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