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%I #14 Jun 11 2023 11:10:12
%S 0,1,13840,4961755,733059110,75073622025,6438673851876,
%T 503519287150295,37463016470769170,2712124797724710645,
%U 193396524783642727120,13675857973300537321251,962624331855762939745950,67586399804656292725004385,4738724382451462432861849980
%N Number of permutations of 4 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.
%H Andrew Howroyd, <a href="/A151642/b151642.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (210, -17985, 836310, -23627805, 429628026, -5189886205, 42366601950, -235447933875, 889918833750, -2267731621875, 3835990781250, -4208760859375, 2865761718750, -1098193359375, 180878906250).
%F a(n) = 70^n - (4*n + 1)*35^n + binomial(4*n+1, 2)*15^n - binomial(4*n+1, 3)*5^n + binomial(4*n+1, 4). - _Andrew Howroyd_, May 07 2020
%F From _G. C. Greubel_, Sep 09 2022: (Start)
%F a(n) = Sum_{j=0..4} (-1)^j*binomial(4*n+1, j)*binomial(8-j, 4)^n.
%F G.f.: x^2*(1 +13630*x +2073340*x^2 -60833350*x^3 -1182529995*x^4 +34295189100*x^5 -173276304000*x^6 -651083647500*x^7 +5378182646875*x^8 -9980105906250*x^9 -2825648437500*x^10 +19397519531250*x^11 +3165380859375*x^12)/( Product_{j=0..4} (1 - binomial(j+4,4)*x)^(5-j) ).
%F E.g.f.: exp(70*x) -(1+140*x)*exp(35*x) +150*x*(1+12*x)*exp(15*x) -(50/3)*x*(3 +48*x +80*x^2)*exp(5*x) +(1/3)*x*(15 +174*x +176*x^2 +32*x^3)*exp(x). (End)
%t Table[Sum[(-1)^j*Binomial[4*n+1,j]*Binomial[8-j,4]^n, {j,0,4}], {n, 30}] (* _G. C. Greubel_, Sep 09 2022 *)
%o (PARI) a(n) = {70^n - (4*n + 1)*35^n + binomial(4*n+1, 2)*15^n - binomial(4*n+1, 3)*5^n + binomial(4*n+1, 4)} \\ _Andrew Howroyd_, May 07 2020
%o (Magma) [(&+[(-1)^j*Binomial(4*n+1,j)*Binomial(8-j,4)^n: j in [0..4]]): n in [1..30]]; // _G. C. Greubel_, Sep 09 2022
%o (SageMath)
%o def A151642(n): return sum((-1)^j*binomial(4*n+1, j)*binomial(8-j, 4)^n for j in (0..4))
%o [A151642(n) for n in (1..30)] # _G. C. Greubel_, Sep 09 2022
%Y Column k=4 of A236463.
%K nonn
%O 1,3
%A _R. H. Hardin_, May 29 2009
%E Terms a(8) and beyond from _Andrew Howroyd_, May 07 2020
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