A376631 -id:A376631 - cp's OEIS frontend

A376660 Decimal expansion of a constant related to the asymptotics of A376630 and A376631.

2, 0, 4, 5, 3, 9, 0, 6, 9, 1, 8, 5, 2, 0, 5, 0, 6, 3, 9, 8, 9, 3, 7, 0, 4, 2, 4, 4, 3, 4, 2, 6, 0, 1, 2, 5, 2, 2, 6, 5, 9, 4, 8, 7, 9, 3, 4, 6, 7, 8, 3, 3, 1, 8, 7, 9, 9, 4, 6, 6, 2, 8, 7, 0, 9, 3, 4, 4, 5, 5, 6, 1, 7, 3, 3, 7, 1, 1, 0, 7, 1, 3, 9, 6, 9, 8, 9, 2, 2, 1, 6, 4, 8, 1, 4, 2, 5, 3, 9, 5, 2, 5, 2, 8, 0, 9

Offset: 1

Vaclav Kotesovec, Oct 01 2024

			2.045390691852050639893704244342601252265948793467833187994662870934455617...

Cf. A333198, A376621, A376630, A376631, A376658, A376659, A376815.

Cf. A088559.

RealDigits[E^Sqrt[3*Log[r]^2/4 + 2*PolyLog[2, r^(1/2)] - Pi^2/6] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Equals limit_{n->infinity} A376630(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376631(n)^(1/sqrt(n)).
Equals A376815^(1/2). - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2/4 + 2*polylog(2, r^(1/2)) - Pi^2/6)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

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

Original entry on oeis.org

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).

A376627 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j))^2.

Original entry on oeis.org

1, 1, 0, 3, 0, 3, 1, 3, 2, 4, 4, 3, 8, 2, 10, 2, 14, 2, 19, 3, 20, 7, 23, 11, 26, 17, 25, 26, 27, 35, 29, 48, 27, 64, 28, 81, 30, 98, 32, 119, 37, 139, 47, 159, 59, 183, 77, 199, 105, 217, 137, 237, 180, 251, 232, 266, 292, 281, 364, 293, 447, 309, 540, 331, 645, 350

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

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

Cf. A179049, A376631, A376659.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*(1 + x^(2*k))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

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

a(n) ~ c * A376659^sqrt(n) / sqrt(n), where c = sqrt(5/168 + sqrt(11/23) * cosh(arccosh(17*sqrt(23)/(2*11^(3/2)))/3)/21) = 0.2512284115765342169430117...

A376632 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 3, 2, 3, 3, 2, 4, 3, 4, 2, 4, 2, 5, 3, 5, 2, 5, 3, 5, 4, 4, 5, 5, 5, 5, 6, 4, 7, 4, 7, 4, 8, 4, 8, 5, 8, 6, 8, 6, 9, 7, 8, 8, 8, 9, 8, 10

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

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

Cf. A306734, A376627, A376631.

nmax = 100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ sqrt(1 + 3/sqrt(5)) * exp(Pi*sqrt(n/30)) / (4*sqrt(n)).

cp's OEIS Frontend

A376660 Decimal expansion of a constant related to the asymptotics of A376630 and A376631.

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

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A376627 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j))^2.

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A376632 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j)).

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cp's OEIS Frontend

A376660 Decimal expansion of a constant related to the asymptotics of A376630 and A376631.

Views

Author

Keywords

Examples

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Mathematica

Formula

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

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A376627 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j))^2.

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A376632 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j)).

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A376627 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j))^2.