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A151634 Number of permutations of 3 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.

%I A151634 #15 Jun 11 2023 11:54:52
%S A151634 0,0,405,128124,12750255,789300477,38464072830,1641724670475,
%T A151634 64856779908606,2445752640197970,89642032274378115,
%U A151634 3228334377697738350,115003717118946936945,4069184219056622926539,143377786266629066071740,5038841894823365860640997,176801555321207696717476200
%N A151634 Number of permutations of 3 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.
%H A151634 Andrew Howroyd, <a href="/A151634/b151634.txt">Table of n, a(n) for n = 1..200</a>
%H A151634 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (126, -6741, 203286, -3863391, 48979386, -427502471, 2613017466, -11265590916, 34232982136, -72719412480, 106245417600, -103853184000, 64584960000, -23040000000, 3584000000).
%F A151634 a(n) = 35^n - (3*n + 1)*20^n + binomial(3*n+1, 2)*10^n - binomial(3*n+1, 3)*4^n + binomial(3*n+1, 4). - _Andrew Howroyd_, May 07 2020
%F A151634 a(n) = Sum_{j=0..6} (-1)^j*binomial(3*n+1, 6-j)*(binomial(j+1, 3))^n. - _G. C. Greubel_, Mar 26 2022
%t A151634 T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*Binomial[3*n+1, k-j+2]*(Binomial[j+1,3])^n, {j, 0, k+2}];
%t A151634 Table[T[n, 4], {n, 30}] (* _G. C. Greubel_, Mar 26 2022 *)
%o A151634 (PARI) a(n) = {35^n - (3*n + 1)*20^n + binomial(3*n+1, 2)*10^n - binomial(3*n+1, 3)*4^n + binomial(3*n+1, 4)} \\ _Andrew Howroyd_, May 07 2020
%o A151634 (Sage)
%o A151634 @CachedFunction
%o A151634 def T(n, k): return sum( (-1)^(k-j)*binomial(3*n+1, k-j+2)*(binomial(j+1, 3))^n for j in (0..k+2) )
%o A151634 [T(n, 4) for n in (1..30)] # _G. C. Greubel_, Mar 26 2022
%Y A151634 Column k=4 of A174266.
%K A151634 nonn
%O A151634 1,3
%A A151634 _R. H. Hardin_, May 29 2009
%E A151634 Terms a(9) and beyond from _Andrew Howroyd_, May 07 2020