A266820 -id:A266820 - cp's OEIS frontend

A266819 Expansion of Product_{k>=1} ((1 + x^k) * (1 + 2*x^k)).

1, 3, 5, 12, 20, 33, 60, 93, 144, 222, 340, 498, 729, 1050, 1486, 2115, 2946, 4068, 5592, 7608, 10278, 13854, 18483, 24528, 32426, 42594, 55677, 72498, 94008, 121290, 156002, 199842, 255012, 324438, 411318, 519771, 655128, 823056, 1031148, 1288590, 1605945

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

Convolution of A000009 and A032302.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000009, A032302, A266820, A266822.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(6*Pi)*n^(3/4)), where c = Pi^2/4 + log(2)^2/2 + polylog(2, -1/2) = 2.259213400307794164599109607216595948859... .

A266822 Expansion of Product_{k>=1} ((1 + x^k) * (1 + 3*x^k)).

Original entry on oeis.org

1, 4, 7, 20, 35, 60, 124, 200, 324, 524, 865, 1320, 2016, 3036, 4453, 6684, 9668, 13856, 19792, 27876, 38956, 54640, 75320, 103268, 141191, 191320, 257892, 346164, 463284, 615292, 814883, 1074556, 1409904, 1844284, 2402756, 3118020, 4038164, 5207344, 6694116

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

Convolution of A000009 and A032308.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000009, A032308, A266819, A266820.

nmax = 40; CoefficientList[Series[Product[(1+x^k) * (1+3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(2*Pi) * n^(3/4)), where c = Pi^2/4 + log(3)^2/2 + polylog(2, -1/3) = 2.761842454190822171313479302500904035832... .

cp's OEIS Frontend

A266819 Expansion of Product_{k>=1} ((1 + x^k) * (1 + 2*x^k)).

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