A280962 Number of integer partitions of the n-th even number or the n-th odd number using predecessors of prime numbers.

1, 2, 4, 7, 11, 17, 26, 37, 53, 74, 101, 137, 183, 240, 314, 406, 520, 662, 837, 1049, 1311, 1627, 2008, 2469, 3021, 3678, 4466, 5397, 6499, 7804, 9338, 11137, 13251, 15715, 18589, 21938, 25823, 30322, 35535, 41544, 48471, 56448, 65602, 76097, 88128, 101867

Offset

0,2

Comments

a(n) is both the number of integer partitions of even numbers {0, 2, 4, 6, ...} = A005843 using primes minus one {1, 2, 4, 6, ...} = A006093 and the number of integer partitions of odd numbers {1, 3, 5, 7, ...} = A005408 using primes minus one.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

G.f. G(x) satisfies: (1+x)*G(x^2) = Product_{p prime} 1/(1-x^(p-1)).

a(n) = A280954(A005408(n)) = A280954(A005843(n)).

Example

The a(4)=11 partitions of 9 are:
(621),   (6111),
(441),   (4221),   (42111),   (411111),
(22221), (222111), (2211111), (21111111),
(111111111).

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=2, 1,
      b(n, prevprime(i))+`if`(i-1>n, 0, b(n-i+1, i)))
    end:
a:= n-> b(2*n, nextprime(2*n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 12 2017

Mathematica

nn=60; invser=Product[1-x^(Prime[n]-1), {n, PrimePi[2nn-1]}];
Table[SeriesCoefficient[1/invser, {x, 0, n}], {n, 1, 2nn-1, 2}]

Crossrefs

Cf. A005408, A005843, A006093, A280954.

Sequence in context: A289177 A249039 A342492 * A096967 A117276 A035295

Adjacent sequences: A280959 A280960 A280961 * A280963 A280964 A280965

Keyword

nonn

Author

Gus Wiseman, Jan 11 2017

Status

approved