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A151624 Number of permutations of 2 indistinguishable copies of 1..n with exactly 2 adjacent element pairs in decreasing order.

%I A151624 #14 Jun 19 2022 02:18:53
%S A151624 0,1,48,603,5158,37257,247236,1568215,9703890,59226357,358722928,
%T A151624 2163496611,13017647646,78225458401,469740168924,2819689366191,
%U A151624 16922139539626,101545622110989,609314411814024,3656015481903355,21936500845191030,131620291694585721
%N A151624 Number of permutations of 2 indistinguishable copies of 1..n with exactly 2 adjacent element pairs in decreasing order.
%H A151624 Andrew Howroyd, <a href="/A151624/b151624.txt">Table of n, a(n) for n = 1..500</a>
%H A151624 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (15,-84,226,-309,207,-54).
%F A151624 a(n) = 6^n - (2*n + 1)*3^n + n*(2*n + 1). - _Andrew Howroyd_, May 06 2020
%F A151624 From _Colin Barker_, Jul 16 2020: (Start)
%F A151624 G.f.: x^2*(1 + 33*x - 33*x^2 - 81*x^3) / ((1 - x)^3*(1 - 3*x)^2*(1 - 6*x)).
%F A151624 a(n) = 15*a(n-1) - 84*a(n-2) + 226*a(n-3) - 309*a(n-4) + 207*a(n-5) - 54*a(n-6) for n>6.
%F A151624 (End)
%F A151624 E.g.f.: x*(3+2*x)*exp(x) - (1+6*x)*exp(3*x) + exp(6*x). - _G. C. Greubel_, Jun 19 2022
%t A151624 Table[6^n -(2*n+1)*3^n +n*(2*n+1), {n,40}] (* _G. C. Greubel_, Jun 19 2022 *)
%o A151624 (PARI) a(n) = {6^n - (2*n + 1)*3^n + n*(2*n + 1)} \\ _Andrew Howroyd_, May 06 2020
%o A151624 (PARI) Vec(x^2*(1 + 33*x - 33*x^2 - 81*x^3) / ((1 - x)^3*(1 - 3*x)^2*(1 - 6*x)) + O(x^25)) \\ _Colin Barker_, Jul 16 2020
%o A151624 (Magma) [(&+[(-1)^j*Binomial(2*n+1,2-j)*Binomial(j+2,2)^n: j in [0..2]]): n in [1..40]]; // _G. C. Greubel_, Jun 19 2022
%o A151624 (SageMath) [6^n -(2*n+1)*3^n +binomial(2*n+1,2) for n in (1..40)] # _G. C. Greubel_, Jun 19 2022
%Y A151624 Column k=2 of A154283.
%K A151624 nonn,easy
%O A151624 1,3
%A A151624 _R. H. Hardin_, May 29 2009
%E A151624 Terms a(12) and beyond from _Andrew Howroyd_, May 06 2020