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

0, 2, 8, 20, 44, 86, 158, 274, 458, 738, 1160, 1778, 2674, 3948, 5744, 8236, 11670, 16344, 22664, 31126, 42390, 57260, 76790, 102260, 135320, 177976, 232778, 302814, 391972, 504948, 647592, 826956, 1051750, 1332438, 1681856, 2115376, 2651726

Offset

0,2

Comments

We set A096586(0) = 0.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

A096586(n) = Sum_k=0^n A093695(k) + 2 * Sum_l=0^(n-1) A000070(l).

Example

a(5) = 2*43 = 86 because:
11 -> 2, 111 -> 12, 12 -> 3, 1111 -> 112, 112 -> 13, 112 -> 22,
13 -> 22, 13 -> 4, 11111 -> 1112, 1112 -> 122, 1112 -> 113, 122 -> 23,
122 -> 113, 113 -> 23, 113 -> 14, 23 -> 14, 14 -> 5,
0 -> 1,
1 -> 11, 1 -> 2, 11 -> 111, 11 -> 12, 2 -> 12, 2 -> 3, 111 -> 1111,
111 -> 112, 12 -> 112, 12 -> 13, 12 -> 22, 3 -> 13, 3 -> 4,
1111 -> 11111, 1111 -> 1112, 112 -> 1112, 112 -> 113, 112 -> 122,
13 -> 113, 13 -> 14, 13 -> 23, 22 -> 23, 22 -> 122, 4 -> 14, 4 -> 5,
which gives 43 transitions and (counting upwards and downwards transitions) we have 2*43 = 86 = A096586(5).

Mathematica

(* 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 *)

Crossrefs

Cf. A093695, A000070.

Sequence in context: A057566 A333642 A009303 * A165751 A131128 A296954

Adjacent sequences: A096583 A096584 A096585 * A096587 A096588 A096589

Keyword

nonn

Author

Thomas Wieder, Jul 02 2004

Extensions

More terms from Robert G. Wilson v, Jul 13 2004

Status

approved