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A280962 Number of integer partitions of the n-th even number or the n-th odd number using predecessors of prime numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A289177 A249039 A342492 * A096967 A117276 A035295
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 11 2017
STATUS
approved

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Last modified March 3 00:46 EST 2024. Contains 370499 sequences. (Running on oeis4.)