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%I #18 Oct 25 2018 17:26:24
%S 1,0,1,1,1,1,3,1,3,4,3,4,7,4,7,10,7,10,15,10,15,21,15,21,29,21,29,39,
%T 29,39,53,39,53,68,53,68,91,68,91,114,91,114,148,114,148,184,148,184,
%U 232,184,232,287,232,287,355,287,355,434,355,434,531,434,531,641,531,641
%N Number of partitions of 2*n into parts that are the average of twin prime pairs (A014574).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = [x^(2*n)] Product_{k>=1} 1/(1 - x^A014574(k)).
%e a(10) = 3 because we have [12, 4, 4], [6, 6, 4, 4] and [4, 4, 4, 4, 4].
%t nmax = 65; Table[CoefficientList[Series[Product[1/(1 - Boole[PrimeQ[k + 1] && PrimeQ[k - 1]] x^k), {k, 1, 2 nmax}], {x, 0, 2 nmax}], x][[n]], {n, 1, 2 nmax + 1, 2}]
%Y Cf. A001097, A014574, A283875.
%K nonn
%O 0,7
%A _Ilya Gutkovskiy_, Oct 25 2018