A376530 -id:A376530 - cp's OEIS frontend

A376152 Decimal expansion of a constant related to the asymptotics of A376530.

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

Offset: 1

Vaclav Kotesovec, Oct 09 2024

			4.988020876600903805335224460790773050832038156091687962387444991919...

Cf. A376530, A376621.

RealDigits[E^(2*Sqrt[Log[r]^2 + 2*PolyLog[2, 1-r] - 2*PolyLog[2, 1-r^3]/3]) /. r -> (-1 - 2/(17 + 3*Sqrt[33])^(1/3) + (17 + 3*Sqrt[33])^(1/3))/3, 10, 120][[1]]

Equals limit_{n->infinity} A376530(n)^(1/sqrt(n)).

Equals exp(2*sqrt(log(r)^2 + 2*polylog(2, 1-r) - 2*polylog(2, 1-r^3)/3)), where r = A192918 = 0.54368901269207636157085597180174... is the real root of the equation r^2 * (1-r^3)^2 = (1-r)^2.

A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 2, 0, 3, 1, 4, 2, 3, 3, 2, 6, 2, 7, 2, 8, 3, 10, 6, 8, 9, 8, 12, 8, 16, 6, 20, 8, 22, 10, 24, 14, 27, 20, 26, 26, 25, 34, 26, 42, 25, 51, 26, 58, 31, 66, 36, 72, 43, 76, 56, 82, 70, 82, 86, 84, 106, 87, 124, 90, 145, 95, 168, 102, 187, 115, 206

Offset: 0

Vaclav Kotesovec, Sep 28 2024

nonn

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

Cf. A216222, A306734, A333179, A340647, A369557, A376530.

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

a(n) ~ A369557(n) / 4.

A376227 a(n) = Product_{k=1..n} (8^k - 1)/(2^k - 1) for n >= 1 with a(0) = 1.

Original entry on oeis.org

1, 7, 147, 10731, 2929563, 3096548091, 12884736606651, 212765655585627963, 13998490777945220569659, 3676801592262757799164923963, 3859174628040582848761303356488763, 16194459027901983959148041623911690081339, 271764285812898926139442499827890355613945218107

Offset: 0

Paul D. Hanna, Oct 09 2024

nonn

			G.f.: A(x) = 1 + 7*x + 147*x^2 + 10731*x^3 + 2929563*x^4 + 3096548091*x^5 + 12884736606651*x^6 + 212765655585627963*x^7 + ...
where the coefficients a(n) of x^n begin
a(0) = 1,
a(1) = 1 * 7,
a(2) = 1 * 7 * 21,
a(3) = 1 * 7 * 21 * 73,
a(4) = 1 * 7 * 21 * 73 * 273,
a(5) = 1 * 7 * 21 * 73 * 273 * 1057,
...

Cf. A376530, A001576, A135576.

{a(n) = (1/3) * prod(k=0,n, 1 + 2^k + 2^(2*k))}
for(n=0,12,print1(a(n),", "))

G.f. A(x) = 1/(1 - 7*x/(1 + 7*x - 21*x/(1 + 21*x - 73*x/(1 + 73*x - 273*x/(1 + 273*x - 1057*x/(1 + 1057*x - 4161*x/(1 + ...))))))), a continued fraction.
a(n) = Product_{k=1..n} (1 + 2^k + 2^(2*k)) for n >= 1 with a(0) = 1.
a(n) = 2^(n*(n+1)/2) * Product_{k=1..n} (1/2^k + 1 + 2^k) for n >= 1.
a(n) ~ c * 2^(n*(n+1)) where c = Product_{n>=1} (1 + 1/2^n + 1/4^n) = 2.975905201850451176749639540825805061981174...

cp's OEIS Frontend

A376152 Decimal expansion of a constant related to the asymptotics of A376530.

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A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

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A376227 a(n) = Product_{k=1..n} (8^k - 1)/(2^k - 1) for n >= 1 with a(0) = 1.

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