A263136 -id:A263136 - cp's OEIS frontend

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

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

A035528 Euler transform of A027656(n-1).

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 9, 13, 19, 28, 42, 57, 84, 115, 164, 227, 313, 429, 588, 799, 1079, 1461, 1952, 2617, 3480, 4627, 6111, 8072, 10604, 13905, 18181, 23701, 30828, 39990, 51763, 66822, 86124, 110687, 142039, 181841, 232409, 296401, 377419, 479635, 608558, 770818

Offset: 0

Christian G. Bower

nonn

Also the weigh transform of A003602. - John Keith, Nov 17 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000219, A003602, A052847, A263150, A263352, A262876, A263136, A263141.

nmax = 50; CoefficientList[Series[-1 + Product[1/(1 - x^(2*k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 100; Flatten[{0, Rest[CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Oct 10 2015 *)

a(n) ~ A^(1/2) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) + Pi^2 * n^(1/3)/(3*2^(8/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(4/3)) / (sqrt(3*Pi) * 2^(71/72) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 02 2015

A262876 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 2, 4, 2, 7, 6, 7, 12, 12, 16, 26, 22, 35, 44, 47, 68, 84, 88, 133, 146, 176, 238, 267, 324, 431, 468, 604, 746, 842, 1068, 1296, 1470, 1884, 2202, 2579, 3220, 3753, 4418, 5483, 6294, 7541, 9144, 10554, 12644, 15191, 17480, 21057, 24896

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

a(n) is the number of partitions of n into parts 3*k-1 of k kinds (k>=1).

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, Oct 12 2015

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
Vaclav Kotesovec, Graph - The asymptotic ratio (70000 terms)

Cf. A000219, A035528, A255610, A262877, A262878, A262879, A262883, A262946, A263136, A263141.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=2, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

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

a(n) ~ Zeta(3)^(19/108) * exp(d1 - Pi^4 / (3888*Zeta(3)) + Pi^2 * n^(1/3) / (2^(4/3)*3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(35/108) * 3^(23/27) * sqrt(Pi) * n^(73/108)), where d1 = A263030 = Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) = -0.188708191979528532376410098649207973592114467268429221509... . - Vaclav Kotesovec, Oct 08 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... .

A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 3, 0, 0, 4, 4, 0, 1, 10, 5, 0, 6, 16, 6, 0, 14, 28, 7, 3, 32, 40, 8, 10, 63, 60, 9, 33, 112, 80, 13, 74, 187, 110, 25, 161, 300, 140, 58, 308, 455, 183, 133, 568, 672, 236, 297, 968, 963, 321, 609, 1609, 1344, 468, 1188, 2546

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. A263139, A263136, A263140, A262878, A263145.

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

a(n) ~ 2^(59/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(20736*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (288*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * 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... .

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

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A035528 Euler transform of A027656(n-1).

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A262876 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^k.

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

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A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.

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

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