A263177 -id:A263177 - cp's OEIS frontend

A263178 Decimal expansion of a constant related to A263141 (negated).

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

Offset: 0

Vaclav Kotesovec, Oct 11 2015

			-0.126995867138823252945270574731135805611834188578689467298972164319...

Cf. A263141, A263177, A263179, A263180, A263181.

NIntegrate[E^(-4*x)/(1-E^(-5*x))^2/x - 1/(25*x^3) - 1/(25*x^2) + 19*E^(-x)/(300*x), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

Integral_{x=0..infinity} exp(-4*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 1/(25*x^2) + 19/(300*x*exp(x)) dx.

A263178 + A263179 + A263180 + A263181 = (log(Gamma(1/5)^3 / ((1+sqrt(5)) * Pi * Gamma(3/5) * 5^(29/12))) - 4*Zeta'(-1))/5 = -0.2745843324986204888923185745... . - Vaclav Kotesovec, Oct 12 2015

A263137 Expansion of Product_{k>=1} 1/(1-x^(4*k-3))^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 3, 3, 6, 9, 9, 9, 13, 19, 23, 23, 28, 42, 51, 56, 62, 84, 108, 120, 133, 170, 219, 253, 276, 335, 427, 503, 556, 650, 815, 977, 1090, 1244, 1525, 1836, 2079, 2344, 2808, 3386, 3876, 4348, 5107, 6121, 7069, 7932, 9176, 10918, 12671, 14257

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A263136, A263143, A263177, A263139.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+4, 4, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100); # after Alois P. Heinz

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^j/(1 - x^(4*j))^2).

a(n) ~ Zeta(3)^(29/288) * exp(d43 - Pi^4/(768*Zeta(3)) + Pi^2 * n^(1/3) / (16*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3)/4) / (2^(77/96) * sqrt(3*Pi) * n^(173/288)), where d43 = A263177 = Integral_{x=0..infinity} exp(-x)/(x*(1 - exp(-4*x))^2) - 1/(16*x^3) - 3/(16*x^2) - 19/(96*x*exp(x)) dx = 0.0960101036186695795680588847641594939129540181270663556962564550198... .

A263176 Decimal expansion of a constant related to A263136 (negated).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Oct 11 2015

			-0.158924147180165035059952001737321408554746599955833696821824808027...

Cf. A263136, A263177.

NIntegrate[E^(-3*x)/(1-E^(-4*x))^2/x - 1/(16*x^3) - 1/(16*x^2) + 5*E^(-x)/(96*x), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

Integral_{x=0..infinity} exp(-3*x)/(x*(1 - exp(-4*x))^2) - 1/(16*x^3) - 1/(16*x^2) + 5/(96*x*exp(x)) dx.

A263176 + A263177 = log(Gamma(1/4))/2 - Zeta'(-1)/4 - 2*log(2)/3 - log(Pi)/4 = -0.062914043561495455491893116973161914641792581828767341125... . - Vaclav Kotesovec, Oct 12 2015

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A263178 Decimal expansion of a constant related to A263141 (negated).

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A263137 Expansion of Product_{k>=1} 1/(1-x^(4*k-3))^k.

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A263176 Decimal expansion of a constant related to A263136 (negated).

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