A255528 -id:A255528 - cp's OEIS frontend

A294846 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(k+1)/2).

1, -1, -2, -4, 0, 3, 17, 24, 40, 9, -24, -149, -250, -435, -395, -281, 514, 1528, 3542, 5127, 6920, 5416, 1368, -11136, -28533, -57051, -82846, -107315, -95655, -43646, 107826, 345877, 727771, 1150968, 1601729, 1766547, 1495154, 183944, -2339567, -6770991, -12701854

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

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Convolution inverse of A028377.

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

Cf. A000217, A000294, A028377, A255528, A284896, A292386.

nmax = 40; CoefficientList[Series[Product[1/(1 + x^k)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

G.f.: Product_{k>=1} 1/(1 + x^k)^A000217(k).

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)^(n/d). - Seiichi Manyama, Nov 14 2017

A261567 Expansion of Product_{k>=1} (1/(1 + 3*x^k))^k.

Original entry on oeis.org

1, -3, 3, -18, 69, -168, 504, -1578, 4800, -14310, 42396, -128049, 385839, -1154271, 3458847, -10386477, 31173873, -93490386, 280426833, -841384614, 2524300014, -7572585150, 22717270491, -68152872885, 204460229394, -613377236379, 1840126774737, -5520391488054

Offset: 0

Vaclav Kotesovec, Aug 24 2015

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In general, for z > 1 or z < -1, if g.f. = Product_{k>=1} (1/(1 - z*x^k))^k, then a(n) ~ c * z^n, where c = Product_{j>=1} 1/(1 - 1/z^j)^(j+1).

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

Cf. A255528, A261566, A261582.

nmax = 40; CoefficientList[Series[Product[(1/(1 + 3*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*3^k/k*x^k/(1-x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

a(n) ~ c * (-3)^n, where c = Product_{j>=1} 1/(1 - 1/(-3)^j)^(j+1) = 0.72392917591300902192520561680114697538581509655711959502191898288595312452...

A284897 Expansion of Product_{k>=1} 1/(1+x^k)^(k^3) in powers of x.

Original entry on oeis.org

1, -1, -7, -20, -8, 99, 455, 958, 715, -3606, -17450, -44157, -61852, 19546, 419786, 1442212, 3084950, 3756436, -2155907, -27112107, -88277693, -187777531, -251308697, -5153980, 1182558343, 4299818445, 9988792754, 16075200671, 12020651310, -29802956283

Offset: 0

Seiichi Manyama, Apr 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..7180

Cf. A248882.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), this sequence (m=3), A284898 (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^3) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A284898 Expansion of Product_{k>=1} 1/(1+x^k)^(k^4) in powers of x.

Original entry on oeis.org

1, -1, -15, -66, -54, 725, 4580, 12739, 3346, -149076, -791226, -2182124, -1656973, 16553206, 100646954, 318795473, 506196578, -818806580, -9148048880, -36415709566, -87180585636, -70923559814, 484810027389, 2992082912770, 9866919438716, 19936695359140

Offset: 0

Seiichi Manyama, Apr 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..3995

Cf. A248883.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), A284897 (m=3), this sequence (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^4) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^4))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284926(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A284899 Expansion of Product_{k>=1} 1/(1+x^k)^(k^5) in powers of x.

Original entry on oeis.org

1, -1, -31, -212, -284, 4935, 43719, 160002, -96747, -4914512, -31358932, -94515285, 97642670, 2823746182, 16834776254, 51617810512, -11233909783, -1137004349695, -7267899354808, -25263858110877, -24537905293857, 319397811973578, 2523465326904492

Offset: 0

Seiichi Manyama, Apr 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..2647

Cf. A248884.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), A284897 (m=3), A284898 (m=4), this sequence (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^5) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^5))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284927(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

Original entry on oeis.org

1, 1, 0, -2, -5, -4, 4, 21, 35, 23, -47, -165, -239, -78, 479, 1273, 1508, -138, -4429, -9451, -8845, 6207, 37937, 67123, 45144, -83355, -308078, -455109, -166872, 873799, 2393041, 2916869, -73472, -8133572, -17828640, -17294146, 10383571, 70275162, 127401305, 90368779, -147825714

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

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N. J. A. Sloane, Transforms

Antidiagonal sums of A279928.

Cf. A067687, A255528, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299211.

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

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

a(0) = 1; a(n) = Sum_{k=1..n} A255528(k-1)*a(n-k).

A303174 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).

Original entry on oeis.org

1, -1, 2, -5, 18, -60, 189, -601, 1967, -6544, 21872, -73247, 246080, -829924, 2808357, -9527485, 32389671, -110316862, 376372802, -1286063899, 4400499380, -15075608840, 51704898623, -177513230200, 610007283817, -2098029341745, 7221561430933, -24875274224531

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

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			a(0) = 1;
a(1) = [x^1] 1/(1 + x) = -1;
a(2) = [x^2] 1/((1 + x)^2*(1 + x^2)) = 2;
a(3) = [x^3] 1/((1 + x)^3*(1 + x^2)^2*(1 + x^3)) = -5;
a(4) = [x^4] 1/((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4)) = 18;
a(5) = [x^5] 1/((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5)) = -60, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + x^k)^(n-k+1) begins:
n = 0: (1),  0,   0,    0,   0,    0,  ...
n = 1:  1, (-1),  1,   -1,   1,   -1,  ...
n = 2:  1,  -2,  (2),  -2,   3,   -4,  ...
n = 3:  1,  -3,   4,  (-5),  9,  -14,  ...
n = 4:  1,  -4,   7,  -10, (18), -30,  ...
n = 5:  1,  -5,  11,  -18,  33, (-60), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A206228, A206229, A255526, A255528, A303173.

Table[SeriesCoefficient[Product[1/(1 + x^k)^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 27}]

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = A318204 = 3.50975432794970334043727352337... and c = 0.2457469629428839220188283... - Vaclav Kotesovec, Aug 21 2018

A261566 Expansion of Product_{k>=1} (1/(1 + 2*x^k))^k.

Original entry on oeis.org

1, -2, 0, -6, 16, -18, 48, -94, 208, -426, 752, -1646, 3360, -6578, 13056, -26358, 53456, -105890, 211392, -424366, 850544, -1699290, 3393136, -6795646, 13601184, -27188130, 54358000, -108752870, 217552976, -435033618, 869999584, -1740145118, 3480497584

Offset: 0

Vaclav Kotesovec, Aug 24 2015

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

Cf. A071109, A255528, A261567.

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

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1 - 1/(-2)^j)^(j+1) = 0.81033497534928929188778847125052151513524786804782471307090750707405...

A279932 Expansion of Product_{k>0} 1/(1 + x^k)^(k*5).

Original entry on oeis.org

1, -5, 5, 0, 30, -51, 5, -130, 220, -125, 649, -605, 870, -2695, 1565, -4852, 7915, -6360, 20625, -17880, 33551, -61015, 50865, -138510, 135485, -224725, 389025, -359610, 849525, -838970, 1417404, -2195205, 2275690, -4756040, 4657940, -8315123, 11174840, -13352315

Offset: 0

Seiichi Manyama, Apr 12 2017

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In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

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

Column k=5 of A279928.

Product_{k>0} 1/(1 + x^k)^(k*m): A027906 (m=-4), A255528 (m=1), A278710 (m=2), A279031 (m=3), A279411 (m=4), this sequence (m=5).

a(n) ~ (-1)^n * exp(-5/12 + 3 * 2^(-5/3) * (5*Zeta(3))^(1/3) * n^(2/3)) * A^5 * (5*Zeta(3))^(1/36) / (2^(5/9) * sqrt(3*Pi) * n^(19/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

G.f.: exp(5*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

Original entry on oeis.org

1, -1, 0, -18, 208, -2400, 36504, -663754, 13808320, -324176418, 8487126400, -245122390601, 7741417124880, -265402847130421, 9816338228638872, -389618889514254225, 16518399076342421248, -745025763154442071130, 35619835529954597786208, -1799459812004380374518790, 95780758238408017088795600

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

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Cf. A255528, A261566, A261567, A266971, A292134, A297326, A298985, A298986, A298987.

Table[SeriesCoefficient[Product[1/(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) ~ (-1)^n * n^n * (1 - 2/n + 6/n^2 - 14/n^3 + 33/n^4 - 70/n^5 + 149/n^6 - 298/n^7 + 591/n^8 - 1132/n^9 + 2139/n^10 + ...), for coefficients, see A005380. - Vaclav Kotesovec, Aug 21 2018

cp's OEIS Frontend

A294846 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(k+1)/2).

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A261567 Expansion of Product_{k>=1} (1/(1 + 3*x^k))^k.

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A284897 Expansion of Product_{k>=1} 1/(1+x^k)^(k^3) in powers of x.

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A284898 Expansion of Product_{k>=1} 1/(1+x^k)^(k^4) in powers of x.

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A284899 Expansion of Product_{k>=1} 1/(1+x^k)^(k^5) in powers of x.

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A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

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A303174 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).

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A261566 Expansion of Product_{k>=1} (1/(1 + 2*x^k))^k.

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A279932 Expansion of Product_{k>0} 1/(1 + x^k)^(k*5).