A229863 The number of integer partitions P of n such that either the length k of P is not a part or P has at least one part equal to 1 (or both).

1, 1, 2, 3, 4, 6, 10, 13, 20, 27, 38, 51, 71, 92, 125, 163, 214, 276, 360, 457, 588, 743, 942, 1182, 1487, 1848, 2306, 2852, 3527, 4335, 5331, 6511, 7958, 9675, 11754, 14223, 17198, 20710, 24928, 29901, 35828, 42808, 51099, 60823, 72333, 85811, 101686, 120244, 142036, 167430, 197170, 231761

Offset

0,3

Links

Table of n, a(n) for n=0..51.

Formula

For n>=2; a(n) = A000041(n) - A008483(n-1)

G.f.: 1/E(x) - x*(1-x)*(1-x^2)/E(x) + x where E(x) = prod(k>=1, 1-x^k ).

Example

a(10) = 38 because there are 42 unrestricted partitions of 10.  All of these except the following four are counted by this sequence: 8+2,5+3+2,4+3+3,4+2+2+2.

Mathematica

nn=51; a=Product[1/(1-x^k), {k, 1, nn}]; b= x(Product[1/(1-x^k), {k, 3, nn}]-1); CoefficientList[Series[a-b, {x, 0, nn}], x]

Prog

(PARI) x='x+O('x^66); Vec( 1/eta(x) - x*(1-x)*(1-x^2)/eta(x) + x )

Crossrefs

Cf. A229816.

Sequence in context: A130126 A288338 A121152 * A215255 A200928 A318558

Adjacent sequences: A229860 A229861 A229862 * A229864 A229865 A229866

Keyword

nonn

Author

Geoffrey Critzer and Joerg Arndt, Oct 01 2013

Status

approved