A258789 -id:A258789 - cp's OEIS frontend

A258788 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)).

1, 1, 3, 12, 47, 192, 811, 3539, 15765, 71362, 327748, 1524081, 7161629, 33958506, 162312471, 781305581, 3784573140, 18435578714, 90261022638, 443956543235, 2192796266004, 10872208762458, 54095648185434, 270029668955605, 1351943521270155, 6787479872751732

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..494

Cf. A258234, A258789, A258797.

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n

Table[SeriesCoefficient[1/Product[x^k*(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}], {x, 0, n*(n+3)/2}], {n, 0, 30}]

a(n) ~ c * d^n / n^2, where d = A258234 = 5.40087190411815415246609111910427005202943771019167057093170601448448... = r^2/(r-1), where r is the root of the equation polylog(2, 1-r) + log(r)^2 = 0, c = 2.578341962163260914344332458898614289944... .

A258268 Decimal expansion of a constant related to A206226.

Original entry on oeis.org

9, 1, 5, 3, 3, 7, 0, 1, 9, 2, 4, 5, 4, 1, 2, 2, 4, 6, 1, 9, 4, 8, 5, 3, 0, 2, 9, 2, 4, 0, 1, 3, 5, 4, 5, 4, 0, 0, 7, 3, 3, 2, 7, 2, 0, 4, 1, 2, 1, 8, 4, 8, 8, 4, 9, 6, 8, 9, 2, 6, 3, 2, 0, 1, 4, 7, 6, 1, 3, 8, 3, 7, 6, 6, 8, 9, 5, 7, 3, 1, 6, 2, 3, 9, 1, 5, 1, 9, 0, 2, 5, 5, 8, 7, 9, 5, 1, 9, 2, 8, 4, 5, 3, 8, 9

Offset: 1

Vaclav Kotesovec, May 25 2015

			9.153370192454122461948530292401354540073...

Cf. A206226, A206227, A206240, A258234, A107379, A173519, A238016, A258789.

r^3/(r-1) /.FindRoot[-PolyLog[2, 1-r] == 3*Log[r]^2/2, {r, E}, WorkingPrecision->120] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A206226(n)^(1/n).
Equals limit n->infinity A206227(n)^(1/n).
Equals limit n->infinity A206240(n)^(1/n).

More digits from Vaclav Kotesovec, Jun 10 2015

A258794 a(n) = [x^n] Product_{k=1..n} 1/(x^(2k)(1-x^k)^3).

Original entry on oeis.org

1, 10, 294, 10592, 433350, 19269768, 910578172, 45070219993, 2313935076132, 122371149279812, 6631958513821919, 366896706349540194, 20656935779581469141, 1180759136663178459661, 68388869189063880001236, 4007252716834400744174729, 237231272998203169561835387

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Cf. A258788, A258789, A258790, A258791, A258792, A258793, A258795, A258796.

Table[SeriesCoefficient[1/Product[x^(2*k)*(1-x^k)^3, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^3, {x, 0, n*(n+2)}], {n, 0, 20}]

a(n) ~ c * d^n / n^3, where d = 70.2047644028747363037741119300640924984352825702388550206966992563459... = r^5/(r-1)^3, where r is the root of the equation polylog(2, 1-r) + (5*log(r)^2)/6 = 0, c = 4.0416205700754156... .

A258791 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)^2).

Original entry on oeis.org

1, 3, 20, 158, 1307, 11352, 102538, 954904, 9112038, 88723163, 878714118, 8829998320, 89848944237, 924291213496, 9600148608620, 100565064076006, 1061498376477423, 11281275452880277, 120635822090127386, 1297256892395670322, 14021436433125959714

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Cf. A258788, A258789, A258793, A258794.

Table[SeriesCoefficient[1/Product[x^k*(1-x^k)^2, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^2, {x, 0, n*(n+3)/2}], {n, 0, 25}]

a(n) ~ c * d^n / n^(5/2), where d = 12.0708016857156441729965623654557363850943928675996965027830903372727... = r^3/(r-1)^2, where r is the root of the equation polylog(2, 1-r) + (3*log(r)^2)/4 = 0, c = 8.36819319541... .

A258793 a(n) = [x^n] Product_{k=1..n} 1/(x^(2k)(1-x^k)^2).

Original entry on oeis.org

1, 4, 55, 896, 16494, 326422, 6812064, 147937628, 3315019979, 76184664934, 1787702723767, 42688437971038, 1034621287862521, 25398832816003228, 630502487733706193, 15805630063826901440, 399669931534045129915, 10184690536676439639278, 261340023300544414822171

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Cf. A258788, A258789, A258791, A258794.

Table[SeriesCoefficient[1/Product[x^(2*k)*(1-x^k)^2, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^2, {x, 0, n*(n+2)}], {n, 0, 20}]

a(n) ~ c * d^n / n^(5/2), where d = 29.1694173246928561008040480794933198469510496062455151175744673506960... = r^4/(r-1)^2, where r is the root of the equation polylog(2, 1-r) + log(r)^2 = 0, c = 2.0036140319464... .

cp's OEIS Frontend

A258788 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)).

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A258268 Decimal expansion of a constant related to A206226.

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A258794 a(n) = [x^n] Product_{k=1..n} 1/(x^(2*k)*(1-x^k)^3).

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A258791 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)^2).

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Mathematica

Formula

A258793 a(n) = [x^n] Product_{k=1..n} 1/(x^(2*k)*(1-x^k)^2).

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Mathematica

Formula

A258794 a(n) = [x^n] Product_{k=1..n} 1/(x^(2k)(1-x^k)^3).

A258793 a(n) = [x^n] Product_{k=1..n} 1/(x^(2k)(1-x^k)^2).