A308986 -id:A308986 - cp's OEIS frontend

A308985 Expansion of Product_{k>=0} (1 + 2*x^(2^k))^2.

1, 4, 8, 16, 24, 32, 48, 64, 88, 96, 128, 128, 176, 192, 256, 256, 344, 352, 448, 384, 512, 512, 640, 512, 688, 704, 896, 768, 1024, 1024, 1280, 1024, 1368, 1376, 1728, 1408, 1856, 1792, 2176, 1536, 2048, 2048, 2560, 2048, 2688, 2560, 3072, 2048, 2736, 2752, 3456

Offset: 0

Ilya Gutkovskiy, Jul 04 2019

nonn

Self-convolution of A001316.

Cf. A000120, A001316, A006046, A007814, A308986.

nmax = 50; CoefficientList[Series[Product[(1 + 2 x^(2^k))^2, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[2^(DigitCount[k, 2, 1] + DigitCount[n - k, 2, 1]), {k, 0, n}]; Table[a[n], {n, 0, 50}]

a(n) = Sum_{k=0..n} 2^(A000120(k)+A000120(n-k)).
a(n) = A001316(n) * Sum_{k=0..n} 2^(A007814(binomial(n,k))).
G.f. A(x) satisfies: A(x) = (1 + 2*x)^2 * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

A309728 G.f. A(x) satisfies: A(x) = A(x^2) / (1 - 2*x).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 132, 264, 558, 1116, 2292, 4584, 9300, 18600, 37464, 74928, 150414, 300828, 602772, 1205544, 2413380, 4826760, 9658104, 19316208, 38641716, 77283432, 154585464, 309170928, 618379320, 1236758640, 2473592208, 4947184416, 9894519246, 19789038492, 39578377812

Offset: 0

Ilya Gutkovskiy, Aug 14 2019

nonn

Cf. A001316, A018819, A171238, A308986.

nmax = 34; A[] = 1; Do[A[x] = A[x^2]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 34; CoefficientList[Series[Product[1/(1 - 2 x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

seq(n)=Vec(1/prod(k=0, logint(n,2), 1 - 2*x^(2^k) + O(x*x^n))) \\ Andrew Howroyd, Aug 14 2019

G.f.: Product_{k>=0} 1/(1 - 2*x^(2^k)).

cp's OEIS Frontend

A308985 Expansion of Product_{k>=0} (1 + 2*x^(2^k))^2.

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