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A301470 Signed recurrence over enriched r-trees: a(n) = (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

%I A301470 #12 May 20 2021 04:24:56
%S A301470 1,0,1,0,1,1,2,3,5,9,15,27,47,87,155,288,524,983,1813,3434,6396,12174,
%T A301470 22891,43810,82925,159432,303559,585966,1121446,2171341,4172932,
%U A301470 8106485,15635332,30445899,58925280,115014681,223210718,436603718,849480835,1664740873
%N A301470 Signed recurrence over enriched r-trees: a(n) = (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.
%H A301470 Alois P. Heinz, <a href="/A301470/b301470.txt">Table of n, a(n) for n = 0..3266</a>
%F A301470 O.g.f.: 1/(1 + x) + x Product_{i > 0} 1/(1 - a(i) x^i).
%F A301470 a(n) = Sum_t (-1)^w(t) where the sum is over all enriched r-trees of size n and w(t) is the sum of leaves of t.
%p A301470 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A301470      `if`(i<1, 0, b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
%p A301470     end:
%p A301470 a:= n-> `if`(n<2, 1-n, b(n-2$2)+b(n-1, n-2)):
%p A301470 seq(a(n), n=0..45);  # _Alois P. Heinz_, Jun 23 2018
%t A301470 a[n_]:=a[n]=(-1)^n+Sum[Times@@a/@y,{y,IntegerPartitions[n-1]}];
%t A301470 Array[a,30]
%t A301470 (* Second program: *)
%t A301470 b[n_, i_] := b[n, i] = If[n == 0, 1,
%t A301470      If[i < 1, 0, b[n, i - 1] + a[i] b[n - i, Min[n - i, i]]]];
%t A301470 a[n_] := If[n < 2, 1 - n, b[n - 2, n - 2] + b[n - 1, n - 2]];
%t A301470 a /@ Range[0, 45] (* _Jean-François Alcover_, May 20 2021, after _Alois P. Heinz_ *)
%Y A301470 Cf. A032305, A055277, A093637, A127524, A196545, A220418, A273866, A273873, A289501, A290261, A290971, A301342-A301345, A301364-A301368, A301422, A301462, A301467, A301469.
%K A301470 nonn
%O A301470 0,7
%A A301470 _Gus Wiseman_, Mar 21 2018