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

Author

Gus Wiseman, Jan 11 2017

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A005408, A005843, A006093, A280954.

Programs

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}]

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)).