A263142 -id:A263142 - cp's OEIS frontend

A263179 Decimal expansion of a constant related to A263142 (negated).

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

Offset: 0

Vaclav Kotesovec, Oct 11 2015

			-0.187803021063745858976409657887070138806758984043175296048481791984...

Cf. A263142, A263178, A263180, A263181.

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

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

A263141 Expansion of Product_{k>=1} 1/(1-x^(5*k-1))^k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 1, 2, 6, 4, 1, 2, 6, 10, 6, 2, 6, 14, 20, 8, 6, 14, 29, 30, 13, 14, 34, 54, 50, 22, 34, 66, 99, 74, 43, 72, 133, 166, 119, 82, 148, 242, 276, 182, 166, 286, 438, 442, 301, 316, 541, 744, 701, 494, 608, 976, 1255

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. A263142, A263143, A263144, A263178, A263145, A035528, A262876, A263136.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=4, 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^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(4*j)/(1 - x^(5*j))^2).

a(n) ~ Zeta(3)^(169/900) * exp(d51 - Pi^4/(10800*Zeta(3))+ Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (300 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(281/900) * 5^(169/450) * sqrt(3*Pi) * n^(619/900)), where d51 = A263178 = 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)) = -0.1269958671388232529452705747311358056... .

A263143 Expansion of Product_{k>=1} 1/(1-x^(5*k-3))^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 2, 4, 2, 7, 2, 7, 6, 7, 12, 7, 16, 12, 16, 26, 16, 35, 22, 40, 44, 40, 68, 47, 80, 84, 86, 133, 94, 167, 146, 182, 238, 198, 314, 274, 358, 431, 386, 593, 494, 696, 754, 761, 1056, 917, 1288, 1326, 1430, 1880, 1664, 2348, 2290

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. A263141, A263142, A263144, A263180, A263147.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=2, 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^(5k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(2*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(2*j)/(1 - x^(5*j))^2).

a(n) ~ Zeta(3)^(121/900) * exp(d53 - Pi^4/(1200*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (100*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(329/900) * 5^(121/450) * sqrt(3*Pi) * n^(571/900)), where d53 = A263180 = Integral_{x=0..infinity} exp(-2*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 3/(25*x^2) - 29/(300*x*exp(x)) = -0.1461681349208040073620067065149936790708... .

A263144 Expansion of Product_{k>=1} 1/(1-x^(5*k-4))^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 6, 9, 9, 9, 9, 13, 19, 23, 23, 23, 28, 42, 51, 56, 56, 62, 84, 108, 120, 126, 133, 170, 219, 253, 268, 283, 335, 427, 503, 547, 574, 658, 815, 977, 1080, 1144, 1265, 1534, 1836, 2068, 2209, 2408, 2832, 3396, 3864, 4178, 4505

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} 1/(1 - x^(s*k-t))^k then a(n) ~ s^(t^2/(3*s^2) - 7/18) * n^(t^2/(6*s^2) - 25/36) * exp(d(s,t) - Pi^4 * t^2 / (432*s^2 * Zeta(3)) + Pi^2 * t * 2^(2/3) * s^(2/3) * n^(1/3) / (12 * s^2 * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / (2^(2/3)*s^(2/3))) / (2^(t^2/(6*s^2) + 11/36) * sqrt(3*Pi) * Zeta(3)^(t^2/(6*s^2) - 7/36)), where d(s,t) = Integral_{x=0..infinity} 1/x * (exp(-(s-t)*x)/(1 - exp(-s*x))^2 - 1/(s^2*x^2) - t/(s^2*x) + exp(-x)*(1/12 - t^2/(2*s^2))) dx.

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. A263141, A263142, A263143, A263181, A263148.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, '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^(5k-4))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^j/(1 - x^(5*j))^2).

a(n) ~ Zeta(3)^(79/900) * exp(d54 - Pi^4/(675*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (75*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(371/900) * 5^(79/450) * sqrt(3*Pi) * n^(529/900)), where d54 = A263181 = Integral_{x=0..infinity} exp(-x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 4/(25*x^2) - 71/(300*x*exp(x)) = 0.1863826906247526303913683646299184833844240863417644... .

A263146 Expansion of Product_{k>=1} (1+x^(5*k-2))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 4, 0, 4, 1, 0, 10, 0, 5, 6, 0, 16, 0, 6, 14, 0, 28, 3, 7, 32, 0, 40, 10, 8, 63, 0, 60, 33, 9, 112, 3, 80, 74, 10, 187, 14, 110, 161, 11, 300, 46, 140, 308, 13, 455, 120, 182, 568, 25, 672, 283, 224, 968, 55, 963

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. A263145, A263147, A263148, A263142, A262879.

nmax = 100; CoefficientList[Series[Product[(1+x^(5k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(3*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(3*j)/(1 - x^(5*j))^2).

a(n) ~ 2^(33/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(8100*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (450*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 3^(4/3) * 2^(2/3) * 5^(1/3) * n^(2/3) / 20) / (30 * sqrt(Pi) * n^(2/3)).

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A263179 Decimal expansion of a constant related to A263142 (negated).

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A263141 Expansion of Product_{k>=1} 1/(1-x^(5*k-1))^k.

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A263143 Expansion of Product_{k>=1} 1/(1-x^(5*k-3))^k.

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A263144 Expansion of Product_{k>=1} 1/(1-x^(5*k-4))^k.

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A263146 Expansion of Product_{k>=1} (1+x^(5*k-2))^k.

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