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%I #12 Jun 11 2023 11:12:56
%S 0,0,1828,21571984,29066972368,16938467955200,6501926870387116,
%T 1978065945844840160,524378714083391626872,127734445724723139679472,
%U 29503552588857666326833140,6587452899587031432766113392,1439127765510353092008927027552,310010313330353917185364216860320
%N Number of permutations of 4 indistinguishable copies of 1..n with exactly 6 adjacent element pairs in decreasing order.
%H Andrew Howroyd, <a href="/A151644/b151644.txt">Table of n, a(n) for n = 1..200</a>
%H G. C. Greubel, <a href="/A151644/a151644.txt">Generating functions and recurrence</a>
%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, signature (924, -378521, 91617024, -14722624611, 1672843754916, -139720634979999, 8802985293454896, -425959267257617574, 16033075393526507736, -473719205662776457290, 11056881154552526304000, -204722830335985725627750, 3014129978676701732565000, -35317513927339931248518750, 329166550817222634139500000, -2435724534109739934786328125, 14263842696601765936879687500, -65800836590491435623611328125, 237643329148874488008750000000, -666631524934449548464990234375, 1438281882495029477739257812500, -2357806371916471525008544921875, 2891080516482183085839843750000, -2594002250018865048706054687500, 1646666887498820332031250000000, -698563762562096740722656250000, 177368526026876953125000000000, -20361182834718017578125000000).
%F From _G. C. Greubel_, Sep 12 2022: (Start)
%F a(n) = Sum_{j=0..6} (-1)^j*binomial(4*n+1, j)*binomial(10-j, 4)^n.
%F G.f., e.g.f., and recurrence are in the file "Generating functions and recurrence". (End)
%t Table[Sum[(-1)^j*Binomial[4*n+1, j]*Binomial[10-j, 4]^n, {j,0,6}], {n, 30}] (* _G. C. Greubel_, Sep 12 2022 *)
%o (Magma) [(&+[(-1)^j*Binomial(4*n+1, j)*Binomial(10-j, 4)^n: j in [0..6]]): n in [1..30]]; // _G. C. Greubel_, Sep 12 2022
%o (SageMath)
%o def A151644(n): return sum((-1)^j*binomial(4*n+1, j)*binomial(10-j, 4)^n for j in (0..6))
%o [A151644(n) for n in (1..30)] # _G. C. Greubel_, Sep 12 2022
%Y Column k=6 of A236463.
%K nonn
%O 1,3
%A _R. H. Hardin_, May 29 2009
%E Terms a(8) and beyond from _Andrew Howroyd_, May 06 2020
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