A267005 -id:A267005 - cp's OEIS frontend

A267004 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k)).

1, 2, 5, 12, 24, 50, 97, 184, 331, 606, 1061, 1834, 3125, 5228, 8673, 14250, 23034, 36894, 58750, 92298, 144398, 223994, 344916, 527116, 801295, 1209870, 1816539, 2713956, 4033169, 5961700, 8775236, 12852444, 18742153, 27225316, 39371647, 56743200, 81467211

Offset: 0

Vaclav Kotesovec, Jan 08 2016

nonn

Convolution of A022629 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A022629, A267005, A267007.

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

A268500 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k)).

Original entry on oeis.org

1, 0, 1, 2, 2, 6, 7, 14, 11, 42, 39, 70, 95, 142, 239, 378, 418, 624, 1106, 1200, 2250, 2836, 4166, 4902, 8021, 10410, 14961, 21268, 29477, 36714, 54172, 68358, 95071, 134946, 168035, 254190, 322335, 427338, 541054, 787264, 964969, 1340730, 1748094, 2311386

Offset: 0

Vaclav Kotesovec, Feb 06 2016

nonn

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

Cf. A022629, A267004, A267005, A268498, A268499.

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

A269153 Expansion of Product_{k>=1} ((1 - kx^k) / (1 - 2x^k)).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 33, 62, 130, 264, 554, 1081, 2237, 4483, 8952, 17933, 35921, 71755, 143502, 286713, 573198, 1146540, 2292277, 4584087, 9166802, 18334880, 36668210, 73336840, 146672469, 293348402, 586695560, 1173398119, 2346805311, 4693617598, 9387229673

Offset: 0

Vaclav Kotesovec, Feb 20 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3300

Cf. A070933, A267005, A269144.

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

a(n) ~ c * 2^n, where c = Product_{k>=1} (2^k - k)/(2^k - 1) = 0.27320499481666294779155052256744055231134605935215258251663905...

A269339 Expansion of Product_{k>=1} (1 - k*x^k) / (1 + x^k).

Original entry on oeis.org

1, -2, -1, 0, 2, 6, -3, 12, -13, -2, -15, 2, -65, 44, 37, -90, 134, 26, 334, -270, 66, 18, 774, -1280, -15, -2266, 2627, -352, -3575, -516, -484, 5660, -3629, 21408, -20639, -1228, 15595, 31796, -22214, 55390, -104447, 58958, -160254, 180704, 17402, -103200

Offset: 0

Vaclav Kotesovec, Feb 24 2016

sign

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

Cf. A022661, A081362, A267004, A267005, A268500.

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

A319756 Expansion of Product_{k>=1} (1 - x^k)/(1 - k*x^k).

Original entry on oeis.org

1, 0, 1, 2, 5, 6, 18, 20, 52, 76, 151, 214, 486, 638, 1265, 1990, 3572, 5288, 9968, 14568, 26270, 40246, 68326, 104414, 182191, 271892, 457636, 708012, 1164554, 1774422, 2945077, 4450020, 7261298, 11138514, 17827308, 27228060, 43860232, 66305840, 105486224, 161284674, 253846152

Offset: 0

Ilya Gutkovskiy, Sep 27 2018

nonn

Convolution of A006906 and A010815.

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A006906, A010815, A267005.

a:=series(mul((1-x^k)/(1-k*x^k),k=1..100),x=0,41): seq(coeff(a,x,n),n=0..40); # Paolo P. Lava, Apr 02 2019

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

x='x+O('x^40); Vec(prod(n=1, 40, (1-x^n)/(1-n*x^n))) \\ Altug Alkan, Sep 27 2018

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(d^(k/d) - 1) ) * x^k/k).
From Vaclav Kotesovec, Sep 27 2018: (Start)
a(n) ~ c * 3^(n/3), where
c = 4613.026226899587659466790384528262900057997961519... if mod(n,3)=0
c = 4612.491093385908314202944836907761153110706939289... if mod(n,3)=1
c = 4612.543916007416515763773288072302642108310934844... if mod(n,3)=2
(End)

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A267004 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k)).

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A268500 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k)).

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A269153 Expansion of Product_{k>=1} ((1 - k*x^k) / (1 - 2*x^k)).

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A269339 Expansion of Product_{k>=1} (1 - k*x^k) / (1 + x^k).

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A319756 Expansion of Product_{k>=1} (1 - x^k)/(1 - k*x^k).

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A269153 Expansion of Product_{k>=1} ((1 - kx^k) / (1 - 2x^k)).