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A271743 Number of set partitions of [n] such that 4 is the largest element of the last block.

%I A271743 #17 Sep 08 2022 08:46:16
%S A271743 10,15,29,63,149,375,989,2703,7589,21735,63149,185343,547829,1627095,
%T A271743 4848509,14479983,43308869,129664455,388469069,1164358623,3490978709,
%U A271743 10468741815,31397836829,94176733263,282496645349,847422827175,2542134263789,7626134355903
%N A271743 Number of set partitions of [n] such that 4 is the largest element of the last block.
%H A271743 Alois P. Heinz, <a href="/A271743/b271743.txt">Table of n, a(n) for n = 4..1000</a>
%H A271743 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%H A271743 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F A271743 G.f.: x^4*(3*x-2)*(2*x^2-15*x+5)/Product_{j=1..3} (j*x-1).
%F A271743 From _Colin Barker_, May 21 2017: (Start)
%F A271743 G.f.: x^4*(2 - 3*x)*(5 - 15*x + 2*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
%F A271743 a(n) = 4 + 2^(n-2) + 3^(n-4) for n>4.
%F A271743 a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>7.
%F A271743 (End)
%e A271743 a(4) = 10: 1234, 123|4, 12|34, 12|3|4, 13|24, 13|2|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
%t A271743 Join[{10},LinearRecurrence[{6,-11,6},{15,29,63},30]] (* _Vincenzo Librandi_, Apr 13 2016 *)
%o A271743 (Magma) I:=[10,15,29,63]; [n le 4 select I[n] else 6*Self(n-1) -11*Self(n-2)+6*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Apr 13 2016
%o A271743 (PARI) Vec(x^4*(2 - 3*x)*(5 - 15*x + 2*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^40)) \\ _Colin Barker_, May 21 2017
%Y A271743 Column k=4 of A271466.
%K A271743 nonn,easy
%O A271743 4,1
%A A271743 _Alois P. Heinz_, Apr 13 2016