A376581 -id:A376581 - cp's OEIS frontend

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 3, 3, 3, 4, 4, 5, 5, 7, 9, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 19, 24, 23, 25, 27, 28, 31, 32, 33, 37, 40, 42, 44, 47, 52, 54, 59, 62, 67, 75, 75, 80, 87, 90, 95, 102, 109, 114, 119, 127, 134, 142, 150, 159, 171, 178, 187, 199, 211

Offset: 0

Vaclav Kotesovec, Sep 29 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A053258, A216222, A306734, A369557, A376542, A376581, A376621.

nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.390989767113799449629...
a(n) ~ c * A376542(n), where c = (108 + 12*sqrt(93))^(1/3)/3 - 4/(108 + 12*sqrt(93))^(1/3) = 1.364655607... is the real root of the equation c*(4 + c^2) = 8.
a(n) ~ c * A369557(n), where c = A347178 = -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3) / sqrt(3) = 0.3411639019... is the real root of the equation 2*c*(1 + 4*c^2) = 1.
a(n) ~ A376631(n) * (A376621/A376660)^sqrt(n).

A376624 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j-1))^2.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 13, 18, 23, 33, 44, 57, 77, 99, 125, 163, 207, 259, 328, 407, 503, 626, 769, 938, 1149, 1397, 1687, 2044, 2458, 2943, 3531, 4213, 5011, 5957, 7055, 8334, 9838, 11580, 13594, 15948, 18661, 21790, 25425, 29593, 34386, 39918, 46250, 53501, 61824, 71325

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A053282, A072223, A376542, A376622, A376581, A376658.

nmax=60; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

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

a(n) ~ (r^(3/4)/sqrt(8*(1 + 3*r^2))) * A376658^sqrt(n) / sqrt(n), where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.

A376622 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1))^2.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 8, 10, 14, 18, 22, 30, 38, 46, 60, 74, 90, 114, 138, 167, 206, 248, 298, 360, 430, 511, 612, 726, 854, 1014, 1192, 1396, 1644, 1918, 2236, 2610, 3032, 3516, 4076, 4714, 5436, 6274, 7220, 8288, 9522, 10906, 12476, 14270, 16282, 18556, 21138, 24038

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A376542, A376581, A376624, A376628.

nmax=60; CoefficientList[Series[Sum[x^(k*(k+1))/Product[1-x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

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

a(n) ~ (5 - sqrt(5)) * exp(Pi*sqrt(2*n/5)) / (8*5^(3/4)*sqrt(n)).

cp's OEIS Frontend

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

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A376624 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j-1))^2.

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A376622 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1))^2.

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cp's OEIS Frontend

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

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A376624 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j-1))^2.

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A376622 G.f.: Sum_{k>=0} x^(k*(k+1)) / Product_{j=1..k} (1 - x^(2*j-1))^2.

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A376624 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j-1))^2.

A376622 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1))^2.