This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A151652 #12 Apr 25 2024 09:19:38
%S A151652 0,400,233300,35138736,3656408776,333546436832,28929582376716,
%T A151652 2460253392551584,207633993378572384,17472055390759340592,
%U A151652 1468614660871788474628,123393355120299525767120,10365952122947333091123000,870767651291136123846763456,73145318723749749228550353212
%N A151652 Number of permutations of 6 indistinguishable copies of 1..n with exactly 3 adjacent element pairs in decreasing order.
%H A151652 Andrew Howroyd, <a href="/A151652/b151652.txt">Table of n, a(n) for n = 1..200</a>
%H A151652 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (165,-9225,237297,-3097935,21760767,-83084547, 168886683,-184022244,101917648,-22588608).
%F A151652 a(n) = 84^n - (6*n + 1)*28^n + binomial(6*n+1, 2)*7^n - binomial(6*n+1, 3). - _Andrew Howroyd_, May 07 2020
%F A151652 From _G. C. Greubel_, Sep 12 2022: (Start)
%F A151652 a(n) = Sum_{j=0..3} (-1)^j*binomial(6*n+1, j)*binomial(9-j, 6)^n.
%F A151652 G.f.: 4*x^2*(100 + 41825*x + 83559*x^2 - 21052241*x^3 + 67903073*x^4 + 129637536*x^5 - 671540492*x^6 - 210827008*x^7)/( Product_{j=0..3} (1 - binomial(j+6, 6)*x)^(4-j) ).
%F A151652 E.g.f.: exp(84*x) - (1 + 168*x)*exp(28*x) + 147*x*(1 + 6*x)*exp(7*x) - x*(35 + 108*x + 36*x^2)*exp(x). (End)
%t A151652 Table[Sum[(-1)^j*Binomial[6*n+1, j]*Binomial[9-j, 6]^n, {j,0,3}], {n, 30}] (* _G. C. Greubel_, Sep 12 2022 *)
%o A151652 (PARI) a(n) = {84^n - (6*n + 1)*28^n + binomial(6*n+1, 2)*7^n - binomial(6*n+1, 3)} \\ _Andrew Howroyd_, May 07 2020
%o A151652 (Magma) [(&+[(-1)^j*Binomial(6*n+1, j)*Binomial(9-j, 6)^n: j in [0..3]]): n in [1..30]]; // _G. C. Greubel_, Sep 12 2022
%o A151652 (SageMath)
%o A151652 def A151652(n): return sum((-1)^j*binomial(6*n+1, j)*binomial(9-j, 6)^n for j in (0..3))
%o A151652 [A151652(n) for n in (1..30)] # _G. C. Greubel_, Sep 12 2022
%Y A151652 Column k=3 of A237252.
%K A151652 nonn
%O A151652 1,2
%A A151652 _R. H. Hardin_, May 29 2009
%E A151652 Terms a(7) and beyond from _Andrew Howroyd_, May 07 2020