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A096586 Number of one-element transitions among all integer partitions of the integers from m=0 to m=n in the unlabeled case.

%I A096586 #10 Mar 28 2015 18:01:35
%S A096586 0,2,8,20,44,86,158,274,458,738,1160,1778,2674,3948,5744,8236,11670,
%T A096586 16344,22664,31126,42390,57260,76790,102260,135320,177976,232778,
%U A096586 302814,391972,504948,647592,826956,1051750,1332438,1681856,2115376,2651726
%N A096586 Number of one-element transitions among all integer partitions of the integers from m=0 to m=n in the unlabeled case.
%C A096586 We set A096586(0) = 0.
%H A096586 Alois P. Heinz, <a href="/A096586/b096586.txt">Table of n, a(n) for n = 0..1000</a>
%F A096586 A096586(n) = Sum_k=0^n A093695(k) + 2 * Sum_l=0^(n-1) A000070(l).
%e A096586 a(5) = 2*43 = 86 because:
%e A096586 11 -> 2, 111 -> 12, 12 -> 3, 1111 -> 112, 112 -> 13, 112 -> 22,
%e A096586 13 -> 22, 13 -> 4, 11111 -> 1112, 1112 -> 122, 1112 -> 113, 122 -> 23,
%e A096586 122 -> 113, 113 -> 23, 113 -> 14, 23 -> 14, 14 -> 5,
%e A096586 0 -> 1,
%e A096586 1 -> 11, 1 -> 2, 11 -> 111, 11 -> 12, 2 -> 12, 2 -> 3, 111 -> 1111,
%e A096586 111 -> 112, 12 -> 112, 12 -> 13, 12 -> 22, 3 -> 13, 3 -> 4,
%e A096586 1111 -> 11111, 1111 -> 1112, 112 -> 1112, 112 -> 113, 112 -> 122,
%e A096586 13 -> 113, 13 -> 14, 13 -> 23, 22 -> 23, 22 -> 122, 4 -> 14, 4 -> 5,
%e A096586 which gives 43 transitions and (counting upwards and downwards transitions) we have 2*43 = 86 = A096586(5).
%t A096586 (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) a[0] = 0; a[n_] := Block[{p = Partitions[n + 1], l = PartitionsP[n + 1]}, Sum[ Length[ Union[ p[[k]] ]]^2 - Length[ Union[ p[[k]] ]], {k, l}]]; b = CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x]; f[n_] := Sum[a[k] + 2b[[k]], {k, n}] - 1; Table[ f[n], {n, 36}] (* _Robert G. Wilson v_, Jul 13 2004 *)
%Y A096586 Cf. A093695, A000070.
%K A096586 nonn
%O A096586 0,2
%A A096586 _Thomas Wieder_, Jul 02 2004
%E A096586 More terms from _Robert G. Wilson v_, Jul 13 2004