A262877 -id:A262877 - cp's OEIS frontend

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

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.

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

A262878 Expansion of Product_{k>=1} (1+x^(3*k-1))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 2, 3, 0, 4, 4, 1, 10, 5, 6, 16, 6, 14, 28, 10, 32, 40, 18, 63, 60, 42, 112, 83, 84, 187, 124, 172, 300, 186, 320, 456, 302, 581, 684, 507, 982, 1004, 874, 1624, 1476, 1508, 2566, 2174, 2582, 3981, 3262, 4338, 6002, 4945, 7138, 8947, 7660

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} (1 + x^(s*k-t))^k then a(n) ~ 2^(t^2/(2*s^2) - 3/4) * s^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4 * t^2 / (1296 * s^2 * Zeta(3)) + Pi^2 * t * 2^(1/3) * 3^(2/3) * s^(2/3) * n^(1/3) / (36 * s^2 * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2^(4/3) * s^(2/3)) ) / (3^(1/3) * s * sqrt(Pi) * n^(2/3)). - 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

Cf. A026007, A027346, A035528, A262876, A262877, A262879, A262884, A262948, A263138, A263145.

with(numtheory):
b:= n-> `if`(n<3, n-1, (p-> [0, -r, 2*r, 0, 0, 2*r+1][p]
         )(1+irem(n+3, 6, 'r'))):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), 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+x^(3k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(2*j)/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]

a(n) ~ exp(2^(-4/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(5/3)*3^(8/3) * Zeta(3)^(1/3)) - Pi^4/(11664*Zeta(3))) * Zeta(3)^(1/6) / (2^(25/36) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A262879 Expansion of Product_{k>=1} (1+x^(3*k-2))^k.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 3, 4, 1, 4, 10, 6, 5, 16, 14, 9, 28, 32, 17, 40, 63, 41, 63, 112, 83, 94, 187, 171, 156, 301, 319, 260, 467, 580, 465, 713, 981, 818, 1095, 1627, 1452, 1682, 2584, 2510, 2632, 4047, 4266, 4162, 6181, 7054, 6685, 9396, 11423, 10753, 14132

Offset: 0

Vaclav Kotesovec, Oct 04 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. A026007, A027346, A035528, A262876, A262877, A262878, A262884, A262949.

nmax=100; CoefficientList[Series[Product[(1+x^(3k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]
Clear[a]; a[n_]:=a[n] = If[n==0, 1, Sum[Sum[d*{0, 2*Floor[d/6] + 1, -Floor[d/6] - 1, 0, 2*Floor[d/6] + 2, 0}[[1 + Mod[d, 6]]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}]

a(n) ~ exp(2^(-4/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(2/3) * 3^(8/3) * Zeta(3)^(1/3)) - Pi^4/(2916*Zeta(3))) * Zeta(3)^(1/6) / (2^(19/36) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

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.

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

Original entry on oeis.org

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.

A262883 Expansion of Product_{k>=1} 1/((1-x^(3k-1))(1-x^(3*k-2)))^k.

Original entry on oeis.org

1, 1, 2, 2, 5, 7, 10, 15, 24, 33, 49, 68, 100, 136, 193, 267, 370, 501, 690, 928, 1260, 1687, 2265, 3007, 4006, 5289, 6987, 9163, 12033, 15698, 20469, 26572, 34470, 44510, 57442, 73861, 94852, 121439, 155287, 198007, 252165, 320335, 406396, 514410, 650288

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262876 and A262877.

Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2001 terms 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. A262876, A262877, A262884, A262923.

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

nmax = 50; CoefficientList[Series[Product[1/((1-x^(3*k-1))*(1-x^(3*k-2)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(-1/18 - Pi^4/(864*Zeta(3)) + (3*Zeta(3)/2)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(5/3)*3^(4/3)*Zeta(3)^(1/3))) * A^(2/3) * Gamma(4/3)^(1/3) * Zeta(3)^(7/54) / (2^(11/27) * 3^(79/108) * Pi^(2/3) * n^(17/27)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

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... .

A263406 Expansion of Product_{k>=1} 1/(1-x^(3*k+2))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 2, 4, 1, 6, 5, 2, 10, 7, 6, 19, 9, 14, 29, 14, 28, 46, 23, 53, 66, 43, 95, 99, 76, 158, 143, 141, 256, 217, 247, 403, 326, 432, 617, 509, 720, 935, 801, 1187, 1399, 1281, 1892, 2087, 2047, 2983, 3107, 3272, 4589, 4647

Offset: 0

Vaclav Kotesovec, Oct 17 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A262877, A263030, A263405, A263414, A263415.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d-2, 3)=0, (d-2)/3, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(3*k+2))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 60; CoefficientList[Series[E^Sum[x^(5*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ c * Zeta(3)^(13/108) * exp(-Pi^4/(972*Zeta(3)) - Pi^2 * n^(1/3) / (2^(1/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(41/108) * 3^(20/27) * sqrt(Pi) * n^(67/108)), where c = 3^(1/3) * Gamma(1/3) * exp(A263030) / sqrt(2*Pi) = 1.2763162741536982965216627321306598385267089489...

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A262878 Expansion of Product_{k>=1} (1+x^(3*k-1))^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A262879 Expansion of Product_{k>=1} (1+x^(3*k-2))^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A262883 Expansion of Product_{k>=1} 1/((1-x^(3*k-1))*(1-x^(3*k-2)))^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

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)).

A262883 Expansion of Product_{k>=1} 1/((1-x^(3k-1))(1-x^(3*k-2)))^k.