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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 26 13:59 EDT 2022. Contains 354092 sequences. (Running on oeis4.)