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 A151627 #14 Jun 11 2023 11:45:15
%S A151627 0,0,0,72,27664,2864328,163809288,6727188848,225167210712,
%T A151627 6590156148912,175992170793456,4407169187423736,105396936343707456,
%U A151627 2437638848729751736,55010494951127561400,1219075824289276443744,26652917330108137129544,576864003740129587504224
%N A151627 Number of permutations of 2 indistinguishable copies of 1..n with exactly 5 adjacent element pairs in decreasing order.
%H A151627 Andrew Howroyd, <a href="/A151627/b151627.txt">Table of n, a(n) for n = 1..200</a>
%H A151627 <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (126, -7245, 252750, -5998905, 102950910, -1325335665, 13104159030, -101064788055, 614053303570, -2956203883287, 11303748373482, -34308033838515, 82354980144330, -155288585585115, 227560437228978, -255076375732668, 213622511296680, -128922602864400, 52813189404000, -13115963880000, 1488034800000).
%F A151627 From _G. C. Greubel_, Sep 07 2022: (Start)
%F A151627 a(n) = Sum_{j=0..5} (-1)^j*binomial(2*n+1, j)*binomial(7-j, 2)^n.
%F A151627 G.f.: 8*x^4*(9 + 2324*x - 12462*x^2 - 1858545*x^3 + 34890010*x^4 - 134744022*x^5 - 1875623070*x^6 + 22965673068*x^7 - 95590873845*x^8 + 93562460910*x^9 + 576877450068*x^10 - 2203266593259*x^11 + 2865061552194*x^12 - 347005909980*x^13 - 2472141497400*x^14 + 1471264884000*x^15 + 318864600000*x^16)/((1-x)^6*(1-3*x)^5*(1-6*x)^4*(1-10*x)^3*(1-15*x)^2*(1-21*x)).
%F A151627 E.g.f.: exp(21*x) - (1 + 30*x)*exp(15*x) + 10*x*(3 + 20*x)*exp(10*x) - 6*x*(1 + 24*x + 48*x^2)*exp(6*x) + (9*x^2/2)*(5 + 20*x + 12*x^2)*exp(3*x) - (x^2/30)*(15 + 90*x + 60*x^2 + 8*x^3)*exp(x). (End)
%t A151627 With[{B = Binomial}, Table[Sum[(-1)^j*B[2n+1,j]*B[7-j,2]^n, {j,0,5}], {n, 30}]] (* _G. C. Greubel_, Sep 07 2022 *)
%o A151627 (Magma) [(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(7-j, 2)^n: j in [0..5]]): n in [1..30]]; // _G. C. Greubel_, Sep 07 2022
%o A151627 (SageMath)
%o A151627 @CachedFunction
%o A151627 def A151627(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(7-j, 2)^n for j in (0..5))
%o A151627 [A151627(n) for n in (1..30)] # _G. C. Greubel_, Sep 07 2022
%Y A151627 Column k=5 of A154283.
%K A151627 nonn
%O A151627 1,4
%A A151627 _R. H. Hardin_, May 29 2009
%E A151627 Terms a(11) and beyond from _Andrew Howroyd_, May 06 2020