A285245 -id:A285245 - cp's OEIS frontend

A285243 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2))^k.

1, 1, 1, 1, 5, 5, 5, 5, 17, 26, 26, 26, 58, 94, 94, 94, 190, 298, 352, 352, 608, 896, 1112, 1112, 1752, 2641, 3289, 3559, 5095, 7499, 9227, 10307, 14051, 20111, 25520, 28760, 38843, 53467, 68191, 76831, 102187, 138283, 175543, 202813, 263905, 355220, 445364

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

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

Cf. A285047, A266941, A285241, A285242, A285245.

nmax = 100; CoefficientList[Series[Product[1/(1 - k*x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]

a(n) ~ c * n * 2^(n/4), where
c = 37.4093119651465404809069752821426852731608123... if mod(n,4)=0
c = 37.6275180026872367633343656570058911570800766... if mod(n,4)=1
c = 37.7650387085085950514850376086515488784106690... if mod(n,4)=2
c = 37.4702467422193571732026074780460498930830447... if mod(n,4)=3

A285244 Expansion of Product_{k>=1} (1 + k*x^(k^2)).

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 0, 3, 3, 0, 0, 6, 6, 0, 4, 4, 0, 0, 8, 8, 0, 0, 0, 17, 17, 0, 0, 34, 34, 0, 0, 0, 15, 15, 6, 6, 30, 30, 12, 32, 20, 0, 0, 58, 58, 0, 0, 43, 103, 60, 24, 38, 134, 120, 48, 48, 21, 21, 0, 102, 144, 42, 8, 240, 232, 0, 16, 72, 146, 90, 0

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

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

Cf. A022629, A033461, A285242, A285245.

nmax = 100; CoefficientList[Series[Product[(1 + k*x^(k^2)), {k,1,nmax}], {x,0,nmax}], x]

A332199 Expansion of Product_{i>=1, j>=1} 1/(1 - ix^(ij)).

Original entry on oeis.org

1, 1, 4, 8, 22, 40, 101, 183, 412, 765, 1586, 2899, 5834, 10484, 20199, 36246, 67758, 119837, 219661, 384200, 690164, 1197423, 2114105, 3632088, 6332797, 10779478, 18555115, 31354932, 53385037, 89494901, 150983344, 251284829, 420218575, 694947117, 1152915743, 1894656801

Offset: 0

Seiichi Manyama, Aug 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006171, A006906, A285245, A318415, A333653.

m = 35; CoefficientList[Series[Product[1/(1 - i*x^(i*j)), {i, 1, m}, {j, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 23 2020 *)

N=40; x='x+O('x^N); Vec(1/prod(i=1, N, prod(j=1, N\i, 1-i*x^(i*j))))

N=40; x='x+O('x^N); Vec(1/prod(k=1, N, prod(d=1, k, 1-(k%d==0)*d*x^k)))

G.f.: Product_{k>0} f(q^k) where f(q) = Product_{i>=1} 1/(1 - i*q^i).

G.f.: Product_{k>0} Product_{d|k} 1/(1 - d*x^k).

A302836 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^(k^2)).

Original entry on oeis.org

1, 1, 4, 27, 260, 3150, 46872, 825944, 16810048, 387952668, 10010010100, 285526191874, 8921263237056, 303013028232642, 11116057874586840, 438023675344410000, 18451248777413066768, 827408674110381669305, 39353155876513869320412, 1978708139249503877752798

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Cf. A001156, A124577, A285245, A291583, A292417, A300974.

Table[SeriesCoefficient[Product[1/(1 - n x^k^2), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ n^n * (1 + 1/n^3 + 1/n^6 + 1/n^8 + 1/n^9 + 1/n^11 + 1/n^12 + 1/n^14 + 2/n^15 + 1/n^16 + 1/n^17 + 2/n^18 + 1/n^19 + 1/n^20 + 2/n^21 + 1/n^22 + 2/n^23 + 4/n^24 + 1/n^25 + 2/n^26 + 4/n^27 + 1/n^28 + 2/n^29 + 5/n^30 + ...), for coefficients see A111178. - Vaclav Kotesovec, Apr 13 2018

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A285243 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2))^k.

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A285244 Expansion of Product_{k>=1} (1 + k*x^(k^2)).

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A332199 Expansion of Product_{i>=1, j>=1} 1/(1 - ix^(ij)).

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A302836 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^(k^2)).

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cp's OEIS Frontend

A285243 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2))^k.

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Formula

A285244 Expansion of Product_{k>=1} (1 + k*x^(k^2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A332199 Expansion of Product_{i>=1, j>=1} 1/(1 - i*x^(i*j)).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A302836 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^(k^2)).

Views

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Formula

A332199 Expansion of Product_{i>=1, j>=1} 1/(1 - ix^(ij)).