login
Number of integer partitions of the n-th even number or the n-th odd number using predecessors of prime numbers.
2

%I #14 May 06 2018 00:48:05

%S 1,2,4,7,11,17,26,37,53,74,101,137,183,240,314,406,520,662,837,1049,

%T 1311,1627,2008,2469,3021,3678,4466,5397,6499,7804,9338,11137,13251,

%U 15715,18589,21938,25823,30322,35535,41544,48471,56448,65602,76097,88128,101867

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

%C 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.

%H Alois P. Heinz, <a href="/A280962/b280962.txt">Table of n, a(n) for n = 0..10000</a>

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

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

%e The a(4)=11 partitions of 9 are:

%e (621), (6111),

%e (441), (4221), (42111), (411111),

%e (22221), (222111), (2211111), (21111111),

%e (111111111).

%p b:= proc(n, i) option remember; `if`(n=0 or i=2, 1,

%p b(n, prevprime(i))+`if`(i-1>n, 0, b(n-i+1, i)))

%p end:

%p a:= n-> b(2*n, nextprime(2*n)):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jan 12 2017

%t nn=60;invser=Product[1-x^(Prime[n]-1),{n,PrimePi[2nn-1]}];

%t Table[SeriesCoefficient[1/invser,{x,0,n}],{n,1,2nn-1,2}]

%Y Cf. A005408, A005843, A006093, A280954.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jan 11 2017