A262946 -id:A262946 - cp's OEIS frontend

A263030 Decimal expansion of a constant related to A262876 and A262946 (negated).

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

Offset: 0

Vaclav Kotesovec, Oct 08 2015

			-0.18870819197952853237641009864920797359211446726842922150941743378232...

Cf. A262876, A262877, A262946, A262947, A263031, A075700, A084448, A263406, A263415.

NIntegrate[1/x*(Exp[-2*x]/(1 - Exp[-3*x])^2 - 1/(9*x^2) - 1/(9*x) + Exp[-x]/36), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) dx.

exp(3*(A263030+A263031)) = A^2 * Gamma(1/3) / (3^(11/12) * exp(1/6) * sqrt(2*Pi)), where A = A074962 is the Glaisher-Kinkelin constant.

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

A262947 Expansion of Product_{k>=1} 1/(1-x^(3k-2))^(3k-2).

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 5, 12, 22, 22, 32, 60, 80, 93, 161, 231, 282, 404, 616, 775, 1041, 1535, 2037, 2600, 3708, 5029, 6411, 8710, 11968, 15315, 20189, 27444, 35619, 45939, 61605, 80422, 102932, 135481, 177391, 226263, 293561, 382984, 488826, 626558, 812750

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

A262946(n)/A262947(n) ~ exp(3*(d1-d2)) * Gamma(1/3)^3 / (2*Pi)^(3/2), where d1 = A263030 and d2 = A263031. - Vaclav Kotesovec, Oct 08 2015

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], Sep 30 2015
Vaclav Kotesovec, Graph - The asymptotic ratio (120000 terms)

Cf. A035382, A262877, A262883, A262923, A262946.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=1, 3*r-2, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

nmax=60; CoefficientList[Series[Product[1/((1-x^(3k-2))^(3k-2)),{k,1,nmax}],{x,0,nmax}],x]
nmax=60; CoefficientList[Series[E^Sum[1/j*x^j*(1+2*x^(3*j))/(1-x^(3*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(23/36) * sqrt(Pi) * Zeta(3)^(5/36) * exp(3*d2 + (3/2)^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(11/36) * Gamma(1/3)^2 * n^(23/36)), where d2 = A263031 = Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) = -0.01453742918328403360502029450226209036054149... . - Vaclav Kotesovec, Oct 08 2015

A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2)).

Original entry on oeis.org

1, 1, 3, 3, 10, 15, 27, 44, 79, 128, 211, 331, 549, 843, 1338, 2061, 3195, 4851, 7384, 11104, 16696, 24774, 36817, 54173, 79560, 116067, 168880, 244293, 352480, 506012, 724531, 1032762, 1468271, 2079525, 2937102, 4134399, 5804795, 8124459, 11342952, 15791650

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262946 and A262947.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
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. A262883, A262876, A262877, A262924, A035386, A035382, A262946, A262947.

nmax=60; CoefficientList[Series[Product[1/((1 - x^(3*k-1))^(3*k-1) * (1 - x^(3*k-2))^(3*k-2)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(-1/6 + 3^(2/3)*(Zeta(3)/2)^(1/3) * n^(2/3)) * A^2 * Zeta(3)^(1/9) / (2^(5/18) * 3^(31/36) * sqrt(Pi) * n^(11/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263031 Decimal expansion of a constant related to A262877 and A262947 (negated).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Oct 08 2015

			-0.01453742918328403360502029450226209036054149759346444138152247405534...

Cf. A262876, A262877, A262946, A262947, A263030, A075700, A084448, A263405, A263414.

NIntegrate[1/x*(Exp[-x]/(1 - Exp[-3*x])^2 - 1/(9*x^2) - 2/(9*x) - 5*Exp[-x]/36), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) dx.

exp(3*(A263030+A263031)) = A^2 * Gamma(1/3) / (3^(11/12) * exp(1/6) * sqrt(2*Pi)), where A = A074962 is the Glaisher-Kinkelin constant.

A285131 Expansion of Product_{k>=0} 1/(1-x^(4k+3))^(4k+3).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 6, 7, 0, 10, 21, 11, 15, 42, 61, 36, 70, 150, 150, 124, 278, 441, 375, 468, 909, 1131, 1018, 1581, 2602, 2810, 2947, 4819, 6768, 6980, 8509, 13389, 16788, 17609, 23722, 34720, 40337, 44863, 63128, 85430, 95887, 114037, 159882, 202699, 227087

Offset: 0

Seiichi Manyama, Apr 15 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=0} 1/(1-x^(m*k+m-1))^(m*k+m-1): A262811 (m=2), A262946 (m=3), this sequence (m=4), A285132 (m=5).

Cf. A285213.

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+3))^(4*k+3), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

x='x+O('x^100); Vec(prod(k=0, 100, 1/(1 - x^(4*k + 3))^(4*k + 3))) \\ Indranil Ghosh, Apr 15 2017

a(n) ~ exp(4*c + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(11/72) / (2^(47/72) * sqrt(3) * Gamma(3/4) * n^(47/72)), where c = Integral_{x=0..inf} ((5/(exp(x)*96) + 1/(exp(3*x)*(1 - exp(-4*x))^2) - 1/(16*x^2) - 1/(16*x))/x) dx = -0.158924147180165035059952001737321408554746599955833696821824808... - Vaclav Kotesovec, Apr 15 2017

A285212 Expansion of Product_{k>=0} (1-x^(3k+2))^(3k+2).

Original entry on oeis.org

1, 0, -2, 0, 1, -5, 0, 10, -8, -5, 26, -11, -28, 62, -4, -101, 111, 43, -260, 182, 228, -583, 202, 715, -1155, 25, 1888, -2034, -851, 4286, -3144, -3418, 8895, -3888, -9806, 16848, -2479, -23812, 29519, 5626, -52156, 46930, 30033, -105320, 66001, 90431, -198736

Offset: 0

Seiichi Manyama, Apr 15 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=0} (1-x^(m*k+m-1))^(m*k+m-1): A285069 (m=2), this sequence (m=3), A285213 (m=4), A285214 (m=5).

Cf. A262946.

nmax = 100; CoefficientList[Series[Product[(1-x^(3*k+2))^(3*k+2), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

x='x+O('x^100); Vec(prod(k=0, 100, (1 - x^(3*k + 2))^(3*k + 2))) \\ Indranil Ghosh, Apr 15 2017

A285132 Expansion of Product_{k>=0} 1/(1-x^(5k+4))^(5k+4).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 0, 10, 9, 0, 0, 20, 36, 14, 0, 35, 90, 101, 19, 56, 180, 320, 202, 108, 315, 730, 859, 492, 533, 1390, 2300, 2139, 1354, 2393, 4835, 6475, 5098, 4619, 8813, 14926, 16395, 12982, 15751, 28962, 41162, 40256, 35200, 51731, 85365, 106145

Offset: 0

Seiichi Manyama, Apr 15 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} 1/(1-x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(c*m + 3 * 2^(-2/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/(6*m) + m/36) / (sqrt(3) * Gamma(1 - 1/m) * m^(1/2 - 5/(6*m) + m/36) * n^(1/2 + 1/(6*m) + m/36)), where c = Integral_{x=0..infinity} exp((m+1)*x) / (x*(exp(m*x)-1)^2) + (1/12 - 1/(2*m^2))/(x*exp(x)) - 1/(m^2*x^3) - 1/(m^2*x^2) dx. - Vaclav Kotesovec, Apr 17 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=0} 1/(1-x^(m*k+m-1))^(m*k+m-1): A262811 (m=2), A262946 (m=3), A285131 (m=4), this sequence (m=5).

Cf. A285214.

nmax = 100; CoefficientList[Series[Product[1/(1-x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

x='x+O('x^100); Vec(prod(k=0, 100, 1/(1 - x^(5*k + 4))^(5*k + 4))) \\ Indranil Ghosh, Apr 15 2017

a(n) ~ exp(5*c + 3*2^(-2/3)*5^(-1/3)*Zeta(3)^(1/3)*n^(2/3)) * (2*Zeta(3))^(31/180) / (sqrt(3) * 5^(17/36) * Gamma(4/5) * n^(121/180)), where c = Integral_{x=0..inf} ((19/(exp(x)*300) + 1/(exp(4*x)*(1-exp(-5*x))^2) - 1/(25*x^2) - 1/(25*x))/x) dx = -0.12699586713882325294527057473113580561183418857868946729897216431919... - Vaclav Kotesovec, Apr 15 2017

A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-1))^(-1/(3k-1)).

Original entry on oeis.org

1, 0, 1, 0, 9, 24, 225, 504, 16065, 27216, 1555281, 6123600, 159249321, 779262120, 31816914129, 240363179784, 8207359913025, 66059979227424, 2145292484152545, 19782668403572256, 1015331126023222281, 7961977144683689400, 454920488042137314561

Offset: 0

Seiichi Manyama, Jul 07 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A001822, A035386, A206303, A262946, A362696.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^(3*k-1))^(1/(3*k-1)))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A001822(k) * a(n-k)/(n-k)!.

cp's OEIS Frontend

A263030 Decimal expansion of a constant related to A262876 and A262946 (negated).

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

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A262947 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^(3*k-2).

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A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3*k-1))^(3*k-1) * (1-x^(3*k-2))^(3*k-2)).

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A263031 Decimal expansion of a constant related to A262877 and A262947 (negated).

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A285131 Expansion of Product_{k>=0} 1/(1-x^(4*k+3))^(4*k+3).

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A285212 Expansion of Product_{k>=0} (1-x^(3*k+2))^(3*k+2).

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A285132 Expansion of Product_{k>=0} 1/(1-x^(5*k+4))^(5*k+4).

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A262947 Expansion of Product_{k>=1} 1/(1-x^(3k-2))^(3k-2).

A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2)).

A285131 Expansion of Product_{k>=0} 1/(1-x^(4k+3))^(4k+3).

A285212 Expansion of Product_{k>=0} (1-x^(3k+2))^(3k+2).

A285132 Expansion of Product_{k>=0} 1/(1-x^(5k+4))^(5k+4).

A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-1))^(-1/(3k-1)).