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A271746 Number of set partitions of [n] such that 7 is the largest element of the last block.

%I A271746 #10 Jun 12 2022 12:34:49
%S A271746 406,1145,3627,12521,46299,181265,745107,3195161,14220459,65412065,
%T A271746 309878787,1507297001,7508078619,38208764465,198238593267,
%U A271746 1046593626041,5612793712779,30528112814465,168152752952547,936705967782281,5270538854994939,29919810501018065
%N A271746 Number of set partitions of [n] such that 7 is the largest element of the last block.
%H A271746 Alois P. Heinz, <a href="/A271746/b271746.txt">Table of n, a(n) for n = 7..1000</a>
%H A271746 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%H A271746 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-175,735,-1624,1764,-720).
%F A271746 G.f.: x^7*(720*x^6-122388*x^5+235852*x^4-161681*x^3+50632*x^2-7381*x+406) / Product_{j=1..6} (j*x-1).
%F A271746 From _Colin Barker_, Jan 04 2018: (Start)
%F A271746 a(n) = 32 + 121*2^(n-7) + 155*3^(n-7) + 5*4^(n-5) + 16*5^(n-7) + 6^(n-7) for n>7.
%F A271746 a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6) for n>13.
%F A271746 (End)
%t A271746 LinearRecurrence[{21,-175,735,-1624,1764,-720},{406,1145,3627,12521,46299,181265,745107},30] (* _Harvey P. Dale_, Jun 12 2022 *)
%o A271746 (PARI) Vec(x^7*(406 - 7381*x + 50632*x^2 - 161681*x^3 + 235852*x^4 - 122388*x^5 + 720*x^6) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)) + O(x^40)) \\ _Colin Barker_, Jan 04 2018
%Y A271746 Column k=7 of A271466.
%K A271746 nonn,easy
%O A271746 7,1
%A A271746 _Alois P. Heinz_, Apr 13 2016