A329436 - OEIS

A329436

Expansion of Sum_{k>=1} (-1 + Product_{j>=2} (1 + x^(k*j))).

0, 1, 1, 2, 2, 4, 3, 5, 6, 8, 7, 13, 10, 16, 18, 22, 21, 34, 29, 44, 45, 56, 56, 82, 78, 100, 109, 136, 137, 185, 181, 231, 247, 295, 317, 399, 404, 490, 533, 638, 669, 817, 853, 1020, 1108, 1276, 1371, 1638, 1728, 2017, 2186, 2519, 2702, 3153, 3371, 3885

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OFFSET

1,4

COMMENTS

Inverse Moebius transform of A025147.

Number of uniform (constant multiplicity) partitions of n not containing 1, ranked by the odd terms of A072774. - Gus Wiseman, Dec 01 2023

LINKS

Table of n, a(n) for n=1..56.

FORMULA

G.f.: Sum_{k>=1} A025147(k) * x^k / (1 - x^k).

a(n) = Sum_{d|n} A025147(d).

EXAMPLE

From Gus Wiseman, Dec 01 2023: (Start)

The a(2) = 1 through a(10) = 8 uniform partitions not containing 1:

(2) (3) (4) (5) (6) (7) (8) (9) (10)

(2,2) (3,2) (3,3) (4,3) (4,4) (5,4) (5,5)

(4,2) (5,2) (5,3) (6,3) (6,4)

(2,2,2) (6,2) (7,2) (7,3)

(2,2,2,2) (3,3,3) (8,2)

(4,3,2) (5,3,2)

(3,3,2,2)

(2,2,2,2,2)

(End)

MATHEMATICA

nmax = 56; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j)), {j, 2, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Table[Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&SameQ@@Length/@Split[#]&]], {n, 0, 30}] (* Gus Wiseman, Dec 01 2023 *)

CROSSREFS

The strict case is A025147.

The version allowing 1 is A047966.

The version requiring 1 is A097986.

Cf. A023645, A047968, A072774, A096765, A329435, A329436, A367586.

Sequence in context: A094051 A159268 A058723 * A182577 A357189 A241450

Adjacent sequences: A329433 A329434 A329435 * A329437 A329438 A329439

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Nov 13 2019

STATUS

approved