A258346 -id:A258346 - cp's OEIS frontend

A028377 Expansion of Product_{m>0} (1+q^m)^(m(m+1)/2).

1, 1, 3, 9, 19, 46, 100, 218, 460, 965, 1975, 3993, 7975, 15712, 30650, 59150, 113093, 214300, 402812, 751165, 1390714, 2557004, 4670770, 8479232, 15302657, 27462424, 49021252, 87057783, 153850769, 270614429, 473850031, 826125184, 1434286323, 2480145226

Offset: 0

N. J. A. Sloane

nonn

Convolved with aerated A000294: [1, 0, 2, 0, 4, 0, 10, 0, 26, ...] = A000294. - Gary W. Adamson, Jun 13 2009

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(n+1)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000294. - Gary W. Adamson, Jun 13 2009

Cf. A027999, A258341, A258342, A258343, A258344, A258345, A258346.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i*(i+1)/2, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 03 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[i*(i+1)/2, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)

a(n) ~ 7^(1/8) * exp(2 * 7^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) + 3^(3/2) * 5^(1/2) * Zeta(3) * n^(1/2) / (2 * 7^(1/2) * Pi^2) - 3^(13/4) * 5^(5/4) * Zeta(3)^2 * n^(1/4) / (4 * 7^(5/4) * Pi^5) + 2025 * Zeta(3)^3 / (98*Pi^8)) / (2^(49/24) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(d+1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 28 2018

A258343 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)/6).

Original entry on oeis.org

1, 1, 4, 14, 36, 101, 260, 669, 1669, 4116, 9932, 23636, 55483, 128532, 294422, 667026, 1496232, 3324720, 7323570, 15998749, 34679966, 74622839, 159454379, 338472749, 713956569, 1496950669, 3120663129, 6469901522, 13343153563, 27379250529, 55907749171

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258342, A258344, A258345, A258346.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(binomial(i+2, 3), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2018

nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)/6),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 28 2018

A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

Original entry on oeis.org

1, 2, 7, 24, 65, 184, 487, 1254, 3145, 7706, 18480, 43490, 100692, 229472, 515802, 1144416, 2508948, 5439642, 11671859, 24801738, 52221911, 109013538, 225718717, 463769652, 945915199, 1915895576, 3854803572, 7706786958, 15314564282, 30255672820, 59440488874

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A027998, A028377, A027999, A258342, A258343, A258344, A258345, A258346, A258347.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) + 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

Original entry on oeis.org

1, 0, 2, 6, 13, 32, 69, 160, 344, 760, 1601, 3384, 7022, 14434, 29361, 59140, 118089, 233754, 459293, 895382, 1733904, 3334914, 6374654, 12111632, 22881777, 42993244, 80362496, 149464404, 276657082, 509740278, 935046158, 1707916988, 3106810873, 5629121054

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A027998, A028377, A027999, A258341, A258342, A258343, A258345, A258346, A258348.

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

a(n) ~ 7^(1/8) / (2^(43/24) * 15^(1/8) * n^(5/8)) * exp(-2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) - 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 21, 39, 76, 145, 294, 581, 1169, 2276, 4435, 8494, 16237, 30768, 58221, 109466, 205223, 382658, 710808, 1314091, 2420437, 4439753, 8115645, 14781062, 26833241, 48550863, 87575527, 157480827, 282362462, 504819198, 900058558, 1600424247

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000335, A023872, A258346, A258347, A258348, A258349, A258350, A258351.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n, 3))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 + Zeta(3)/(8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2/(90*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5 * Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

A258342 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)).

Original entry on oeis.org

1, 6, 39, 224, 1131, 5412, 24411, 105078, 435048, 1740312, 6755877, 25533330, 94205738, 340064322, 1203313782, 4180514846, 14279610417, 48013553310, 159086287869, 519912616614, 1677331973910, 5345927500226, 16843574682291, 52494817082952, 161923200857711

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258343, A258344, A258345, A258346, A258350.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^(1/5) * Zeta(5)^(1/10) / (2^(91/120) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401*Pi^16 / (1749600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - Zeta(3)^2 / (25*Zeta(5)) + (343*Pi^12/(405000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (750 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (180000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5) * Zeta(3) / (2^(3/5) * (5*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5)) * n^(3/5) + 5*3^(2/5) * (5*Zeta(5)/2)^(1/5)/4 * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

A258345 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 135, 354, 972, 2684, 6990, 17802, 44627, 111582, 277329, 684164, 1671984, 4050096, 9735209, 23238480, 55120950, 129940442, 304502583, 709464798, 1643920584, 3789158988, 8690016942, 19833550266, 45056952957, 101900481462, 229462378987

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258346, A258351.

nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^(1/5) * Zeta(5)^(1/10) / (2^(91/120) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (1749600000000*Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - Zeta(3)^2 / (25*Zeta(5)) + (-343 * Pi^12 / (405000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (180000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5) * Zeta(3) / (2^(3/5) * (5*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5)) * n^(3/5) + 5*3^(2/5) * ((5*Zeta(5))/2)^(1/5)/4 * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

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A028377 Expansion of Product_{m>0} (1+q^m)^(m(m+1)/2).

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A258343 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)/6).

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A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

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A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

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A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)*(k-2)/6).

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A258342 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)).

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A258345 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)).

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A258343 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)/6).

A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

A258342 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)).

A258345 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)).