A258788 -id:A258788 - cp's OEIS frontend

A258234 Decimal expansion of a constant related to A107379.

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

Offset: 1

Vaclav Kotesovec, May 24 2015

Limit n->infinity (Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx)^(1/n) = Limit n->infinity (A258191(n)/A258192(n))^(1/n) = 1/A258234 = 0.18515528932235959464731321119795428527382236445907508398560553036... .

			5.4008719041181541524660911191042700520294...

StackExchange - Mathematica, No response to an infinite limit

Cf. A107379, A173519, A258191, A258192, A258268, A258788.

r^2/(r-1) /.FindRoot[-PolyLog[2, 1-r] == Log[r]^2, {r, E}, WorkingPrecision->117] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A107379(n)^(1/n).

Equals limit n->infinity A173519(n)^(1/n).

More terms from Vaclav Kotesovec, Jun 09 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... .

A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2k)(1-x^k)).

Original entry on oeis.org

1, 1, 5, 27, 169, 1115, 7760, 55748, 411498, 3101490, 23785645, 185064559, 1457664666, 11602828475, 93205739436, 754751603157, 6155229065861, 50515624923790, 416930705579538, 3458726257239312, 28825340825747729, 241245120218823892, 2026803168946440648

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A258268, A258788, A258791, A258793, A258794.

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

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

a(n) ~ c * d^n / n^2, where d = A258268 = 9.15337019245412246194853029240135454007332720412184884968926320147613... = r^3/(r-1), where r is the root of the equation polylog(2, 1-r) + 3*log(r)^2/2 = 0, c = 0.8069142856822510276258439534144172057548... .

A258790 a(n) = [x^n] Product_{k=1..n} 1/(x^(3k)(1-x^k)).

Original entry on oeis.org

1, 1, 6, 48, 411, 3765, 36308, 363446, 3742085, 39405777, 422669224, 4603472960, 50790334667, 566603884871, 6381702580969, 72481863380510, 829331355150992, 9551576115706329, 110654552651370400, 1288710163262774157, 15080440970246785366, 177237948953055593475

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A258788, A258790, A258792, A258794, A258795, A258796.

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

Table[SeriesCoefficient[1/Product[x^(3*k)*(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}], {x, 0, n*(3*n+5)/2}], {n, 0, 25}]

a(n) ~ c * d^n / n^2, where d = 12.8718984948677835397002665286811919572579479691341210018008114644121... = r^4/(r-1), where r is the root of the equation polylog(2, 1-r) + 2*log(r)^2 = 0, c = 0.44720199058408831652920046766862756... .

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... .

A258792 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)^3).

Original entry on oeis.org

1, 6, 69, 915, 12978, 194688, 3051617, 49526487, 826910754, 14135805042, 246508115583, 4372617452085, 78714369892152, 1435357362134796, 26472477913596486, 493178852479545556, 9270953614684288962, 175695092091980786166, 3354069936616380522256

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Cf. A258788, A258790, A258794, A258795, A258796.

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

a(n) ~ c * d^n / n^3, where d = 22.0610202494679061193859054301626736218023392292898139172609021542610... = r^4/(r-1)^3, where r is the root of the equation polylog(2, 1-r) + (2*log(r)^2)/3 = 0, c = 20.953639522741... .

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... .

A258795 a(n) = [x^n] Product_{k=1..n} 1/(x^(3k)(1-x^k)^2).

Original entry on oeis.org

1, 5, 112, 3216, 104112, 3661517, 136580866, 5323418568, 214685704402, 8897404908604, 377068336570902, 16280261371485594, 714081427614467553, 31747177836376617322, 1428084942303149795972, 64902413675181889657064, 2976483322906106920966911

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Cf. A258788, A258790, A258791, A258792, A258793, A258794, A258796.

Table[SeriesCoefficient[1/Product[x^(3*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*(3*n+5)/2}], {n, 0, 20}]

a(n) ~ c * d^n / n^(5/2), where d = 53.0676066669703028123492951828168330443393201750491213178019371417684... = r^5/(r-1)^2, where r is the root of the equation polylog(2, 1-r) + (5*log(r)^2)/4 = 0, c = 0.983501005499107... .

A258796 a(n) = [x^n] Product_{k=1..n} 1/(x^(3k)(1-x^k)^3).

Original entry on oeis.org

1, 15, 882, 67385, 5938518, 575782833, 59765085601, 6529604684991, 742474127495175, 87176531917206953, 10508492822243329854, 1294860745291809207237, 162553748258042032103013, 20735748733960087597815855, 2682101373558320853655174803

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Cf. A258788, A258790, A258792, A258794, A258795.

Table[SeriesCoefficient[1/Product[x^(3*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*(3*n+5)/2}], {n, 0, 20}]

a(n) ~ c * d^n / n^3, where d = 157.540286488430979726276374519534734829527107090287337321136938826336... = r^6/(r-1)^3, where r is the root of the equation polylog(2, 1-r) + log(r)^2 = 0, c = 1.797864597437050667... .

cp's OEIS Frontend

A258234 Decimal expansion of a constant related to A107379.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2*k)*(1-x^k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A258790 a(n) = [x^n] Product_{k=1..n} 1/(x^(3*k)*(1-x^k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A258792 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)^3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A258795 a(n) = [x^n] Product_{k=1..n} 1/(x^(3*k)*(1-x^k)^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A258796 a(n) = [x^n] Product_{k=1..n} 1/(x^(3*k)*(1-x^k)^3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

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

A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2k)(1-x^k)).

A258790 a(n) = [x^n] Product_{k=1..n} 1/(x^(3k)(1-x^k)).

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

A258795 a(n) = [x^n] Product_{k=1..n} 1/(x^(3k)(1-x^k)^2).

A258796 a(n) = [x^n] Product_{k=1..n} 1/(x^(3k)(1-x^k)^3).