A263143 -id:A263143 - cp's OEIS frontend

A263180 Decimal expansion of a constant related to A263143 (negated).

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

Offset: 0

Vaclav Kotesovec, Oct 11 2015

			-0.146168134920804007362006706514993679070882217048053774943704174890...

Cf. A263143, A263178, A263179, A263181.

NIntegrate[E^(-2*x)/(1-E^(-5*x))^2/x - 1/(25*x^3) - 3/(25*x^2) - 29*E^(-x)/(300*x), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

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)) 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... .

A263142 Expansion of Product_{k>=1} 1/(1-x^(5*k-2))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 3, 2, 1, 6, 2, 5, 6, 2, 11, 6, 7, 15, 6, 21, 15, 12, 30, 15, 34, 35, 22, 58, 35, 59, 70, 43, 108, 76, 95, 142, 85, 187, 157, 161, 263, 174, 318, 307, 274, 480, 336, 534, 583, 479, 836, 649, 879, 1068, 840, 1433, 1211

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, A263143, A263144, A263179, A263146, A262877.

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

a(n) ~ Zeta(3)^(151/900) * exp(d52 - Pi^4/(2700*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (150 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(299/900) * 5^(151/450) * sqrt(3*Pi) * n^(601/900)), where d52 = A263179 = 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)) = -0.187803021063745858976409657887070138806... .

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... .

A263147 Expansion of Product_{k>=1} (1+x^(5*k-3))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 3, 0, 4, 0, 1, 4, 0, 10, 0, 6, 5, 0, 16, 0, 14, 6, 3, 28, 0, 32, 7, 10, 40, 0, 63, 8, 33, 60, 3, 112, 9, 74, 80, 14, 187, 10, 161, 110, 46, 300, 12, 308, 140, 120, 455, 24, 568, 182, 283, 672, 54, 968, 224, 594, 963, 146

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, A263146, A263148, A263143.

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

a(n) ~ 2^(43/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(3600*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (300*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)).

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... .

cp's OEIS Frontend

A263180 Decimal expansion of a constant related to A263143 (negated).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A263142 Expansion of Product_{k>=1} 1/(1-x^(5*k-2))^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A263147 Expansion of Product_{k>=1} (1+x^(5*k-3))^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula