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A271747 Number of set partitions of [n] such that 8 is the largest element of the last block. 2

%I #9 Jan 04 2018 17:29:01

%S 1754,5649,20085,77133,315597,1362669,6164685,29058813,142084077,

%T 717966669,3737612685,19991467293,109605434157,614681711469,

%U 3519553748685,20540447808573,121996580169837,736352527581069,4510823754140685,28011087761890653,176122939449075117

%N Number of set partitions of [n] such that 8 is the largest element of the last block.

%H Alois P. Heinz, <a href="/A271747/b271747.txt">Table of n, a(n) for n = 8..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (28,-322,1960,-6769,13132,-13068,5040).

%F G.f.: x^8 *(5040*x^7 -3145476*x^6 +6799268*x^5 -5424029*x^4 +2104109*x^3 -426701*x^2 +43463*x -1754)/Product_{j=1..7} (j*x-1).

%F From _Colin Barker_, Jan 04 2018: (Start)

%F a(n) = 64 + 91*2^(n-6) + 245*2^(2*n-15) + 11*2^(n-7)*3^(n-8) + 217*3^(n-7) + 161*5^(n-8) + 7^(n-8) for n>8.

%F a(n) = 28*a(n-1) - 322*a(n-2) + 1960*a(n-3) - 6769*a(n-4) + 13132*a(n-5) - 13068*a(n-6) + 5040*a(n-7) for n>15.

%F (End)

%o (PARI) Vec(x^8*(1754 - 43463*x + 426701*x^2 - 2104109*x^3 + 5424029*x^4 - 6799268*x^5 + 3145476*x^6 - 5040*x^7) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^40)) \\ _Colin Barker_, Jan 04 2018

%Y Column k=8 of A271466.

%K nonn,easy

%O 8,1

%A _Alois P. Heinz_, Apr 13 2016

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