cp's OEIS Frontend

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

%I A280962 #14 May 06 2018 00:48:05
%S A280962 1,2,4,7,11,17,26,37,53,74,101,137,183,240,314,406,520,662,837,1049,
%T A280962 1311,1627,2008,2469,3021,3678,4466,5397,6499,7804,9338,11137,13251,
%U A280962 15715,18589,21938,25823,30322,35535,41544,48471,56448,65602,76097,88128,101867
%N A280962 Number of integer partitions of the n-th even number or the n-th odd number using predecessors of prime numbers.
%C A280962 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 A280962 Alois P. Heinz, <a href="/A280962/b280962.txt">Table of n, a(n) for n = 0..10000</a>
%F A280962 G.f. G(x) satisfies: (1+x)*G(x^2) = Product_{p prime} 1/(1-x^(p-1)).
%F A280962 a(n) = A280954(A005408(n)) = A280954(A005843(n)).
%e A280962 The a(4)=11 partitions of 9 are:
%e A280962 (621),   (6111),
%e A280962 (441),   (4221),   (42111),   (411111),
%e A280962 (22221), (222111), (2211111), (21111111),
%e A280962 (111111111).
%p A280962 b:= proc(n, i) option remember; `if`(n=0 or i=2, 1,
%p A280962       b(n, prevprime(i))+`if`(i-1>n, 0, b(n-i+1, i)))
%p A280962     end:
%p A280962 a:= n-> b(2*n, nextprime(2*n)):
%p A280962 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 12 2017
%t A280962 nn=60;invser=Product[1-x^(Prime[n]-1),{n,PrimePi[2nn-1]}];
%t A280962 Table[SeriesCoefficient[1/invser,{x,0,n}],{n,1,2nn-1,2}]
%Y A280962 Cf. A005408, A005843, A006093, A280954.
%K A280962 nonn
%O A280962 0,2
%A A280962 _Gus Wiseman_, Jan 11 2017