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A339277 Number of partitions of 2*n into powers of 2 where every part appears at least 2 times.

%I A339277 #6 Nov 29 2020 22:48:18
%S A339277 1,1,2,3,5,6,9,11,16,19,25,30,39,45,56,65,81,92,111,127,152,171,201,
%T A339277 226,265,295,340,379,435,480,545,601,682,747,839,920,1031,1123,1250,
%U A339277 1361,1513,1640,1811,1963,2164,2335,2561,2762,3027,3253,3548,3813,4153,4448,4827,5167
%N A339277 Number of partitions of 2*n into powers of 2 where every part appears at least 2 times.
%H A339277 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A339277 G.f.: (1/(1 - x^3)) * Product_{k>=0} 1/(1 - x^(2^k)).
%F A339277 G.f.: (1/(1 - x)) * Product_{k>=0} (1 + x^(2^(k+1))/(1 - x^(2^k))).
%F A339277 a(n) = [x^(2*n)] Product_{k>=0} (1 + x^(2^(k+1))/(1 - x^(2^k))).
%F A339277 a(n) = Sum_{k=0..n} U(k,-1/2) * A000123(n-k), where U(k,x) is the Chebyshev U-polynomial.
%e A339277 a(4) = 5 because we have [4, 4], [2, 2, 2, 2], [2, 2, 2, 1, 1], [2, 2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].
%t A339277 nmax = 55; CoefficientList[Series[(1/(1 - x^3)) Product[1/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
%t A339277 A000123[0] = 1; A000123[n_] := A000123[n] = A000123[Floor[n/2]] + A000123[n - 1]; a[n_] := Sum[ChebyshevU[k, -1/2] A000123[n - k], {k, 0, n}]; Table[a[n], {n, 0, 55}]
%Y A339277 Cf. A000079, A000123, A007690, A018819, A339279.
%K A339277 nonn
%O A339277 0,3
%A A339277 _Ilya Gutkovskiy_, Nov 29 2020