A263137 -id:A263137 - cp's OEIS frontend

A263177 Decimal expansion of a constant related to A263137.

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

Offset: 0

Vaclav Kotesovec, Oct 11 2015

			0.0960101036186695795680588847641594939129540181270663556962564550198...

Cf. A263137, A263176.

NIntegrate[E^(-x)/(1-E^(-4*x))^2/x - 1/(16*x^3) - 3/(16*x^2) - 19*E^(-x)/(96*x), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

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.

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

A263136 Expansion of Product_{k>=1} 1/(1-x^(4*k-1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 2, 3, 1, 2, 6, 5, 2, 6, 11, 7, 6, 15, 21, 12, 15, 30, 34, 22, 35, 58, 59, 43, 70, 108, 95, 85, 142, 187, 161, 167, 263, 318, 274, 318, 480, 534, 471, 595, 836, 879, 819, 1081, 1433, 1442, 1429, 1915, 2391, 2365, 2483, 3314, 3947

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. A035528, A262876, A263141, A263137, A263176, A263138.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+4, 4, 'r')=3, 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-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(4*j))^2).

a(n) ~ Zeta(3)^(53/288) * exp(d41 - Pi^4/(6912*Zeta(3)) + Pi^2 * n^(1/3) / (48*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3)/4) / (sqrt(3*Pi) * 2^(101/96) * n^(197/288)), where d41 = A263176 = 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)) = -0.158924147180165035059952001737321408554746599955833696821824808027... .

A263139 Expansion of Product_{k>=1} (1+x^(4*k-3))^k.

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 4, 10, 6, 0, 5, 16, 14, 3, 6, 28, 32, 10, 7, 40, 63, 33, 11, 60, 112, 74, 23, 80, 187, 161, 56, 111, 300, 308, 131, 152, 455, 568, 295, 223, 672, 968, 607, 356, 967, 1609, 1186, 618, 1367, 2546, 2189, 1132, 1926, 3941

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. A263138, A263137, A263147.

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

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

a(n) ~ 2^(83/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(2304*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (96*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * sqrt(Pi) * n^(2/3)).

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A263177 Decimal expansion of a constant related to A263137.

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A263136 Expansion of Product_{k>=1} 1/(1-x^(4*k-1))^k.

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A263139 Expansion of Product_{k>=1} (1+x^(4*k-3))^k.

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