A266822 -id:A266822 - 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... .

A266820 Expansion of Product_{k>=1} ((1 + 2x^k) (1 + 3*x^k)).

Original entry on oeis.org

1, 5, 11, 30, 66, 115, 252, 445, 762, 1350, 2238, 3690, 5909, 9480, 14460, 22475, 34326, 51150, 76398, 111810, 163350, 236610, 339667, 482040, 684060, 960780, 1340953, 1863570, 2573022, 3533310, 4830822, 6580170, 8900382, 12011430, 16125198, 21567965

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

Convolution of A032302 and A032308.

In general, for m1 > 0 and m2 > 0, if g.f. = Product_{k>=1} ((1 + m1*x^k) * (1 + m2*x^k)) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m1+1)*(m2+1)*Pi) * n^(3/4)), where c = Pi^2/3 + log(m1)^2/2 + log(m2)^2/2 + polylog(2, -1/m1) + polylog(2, -1/m2).

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

Cf. A032302, A032308, A266819, A266822.

nmax = 40; CoefficientList[Series[Product[(1+2*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(3*Pi) * n^(3/4)), where c = Pi^2/3 + log(2)^2/2 + log(3)^2/2 + polylog(2, -1/2) + polylog(2, -1/3) = 6.665989921346842772385004076363525173910446415877... .

cp's OEIS Frontend

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

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A266820 Expansion of Product_{k>=1} ((1 + 2x^k) (1 + 3*x^k)).

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cp's OEIS Frontend

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

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A266820 Expansion of Product_{k>=1} ((1 + 2*x^k) * (1 + 3*x^k)).

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A266820 Expansion of Product_{k>=1} ((1 + 2x^k) (1 + 3*x^k)).