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A151640 Number of permutations of 4 indistinguishable copies of 1..n with exactly 2 adjacent element pairs in decreasing order.

%I A151640 #18 Sep 08 2022 05:05:18
%S A151640 0,36,1828,40136,693960,11000300,168594156,2550000528,38371094416,
%T A151640 576250000820,8647558594740,129734375001176,1946130371095128,
%U A151640 29192578125001596,437892028808595580,6568398437500002080,98526072692871096096,1477891601562500002628
%N A151640 Number of permutations of 4 indistinguishable copies of 1..n with exactly 2 adjacent element pairs in decreasing order.
%H A151640 Andrew Howroyd, <a href="/A151640/b151640.txt">Table of n, a(n) for n = 1..500</a>
%H A151640 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (28,-253,976,-1675,1300,-375).
%F A151640 a(n) = 15^n - (4*n + 1)*5^n + 2*n*(4*n + 1). - _Andrew Howroyd_, May 06 2020
%F A151640 From _Colin Barker_, May 06 2020: (Start)
%F A151640 G.f.: 4*x^2*(9 + 205*x - 485*x^2 - 625*x^3) / ((1 - x)^3*(1 - 5*x)^2*(1 - 15*x)).
%F A151640 a(n) = 28*a(n-1) - 253*a(n-2) + 976*a(n-3) - 1675*a(n-4) + 1300*a(n-5) - 375*a(n-6) for n > 6. (End)
%F A151640 E.g.f.: exp(15*x) - (1+20*x)*exp(5*x) + 2*x*(5+4*x)*exp(x). - _G. C. Greubel_, Sep 08 2022
%t A151640 Table[Sum[(-1)^j*Binomial[4*n+1,j]*Binomial[6-j,4]^n, {j,0,2}], {n,30}] (* _G. C. Greubel_, Sep 08 2022 *)
%o A151640 (PARI) a(n) = {15^n - (4*n + 1)*5^n + 2*n*(4*n + 1)} \\ _Andrew Howroyd_, May 06 2020
%o A151640 (PARI) concat(0, Vec(4*x^2*(9 + 205*x - 485*x^2 - 625*x^3) / ((1 - x)^3*(1 - 5*x)^2*(1 - 15*x)) + O(x^20))) \\ _Colin Barker_, May 07 2020
%o A151640 (Magma) [15^n -(4*n+1)*5^n +2*n*(4*n+1): n in [1..30]]; // _G. C. Greubel_, Sep 08 2022
%o A151640 (SageMath) [15^n -(4*n+1)*5^n +2*n*(4*n+1) for n in (1..30)] # _G. C. Greubel_, Sep 08 2022
%Y A151640 Column k=2 of A236463.
%K A151640 nonn,easy
%O A151640 1,2
%A A151640 _R. H. Hardin_, May 29 2009
%E A151640 Terms a(9) and beyond from _Andrew Howroyd_, May 06 2020