cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A366018 Decimal expansion of a constant related to the asymptotics of A181315.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Sep 26 2023

Keywords

Examples

			0.482420439587319764659364391266849418507665645926542970519109122...

Crossrefs

Cf. A181315, A270914.

Programs

  • Mathematica
    RealDigits[Sqrt[s/(Pi*Derivative[0, 2][QPochhammer][-1, r*s])]/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]]

Formula

Equals limit_{n->infinity} A181315(n) * n^(3/2) / A270914^n.

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 25 2016

Keywords

Comments

This constant is very close to exp(5*Pi/(6*sqrt(2))) / sqrt(2) = 4.502476748630924546525119125234175537729... - Vaclav Kotesovec, May 17 2018

Examples

			4.502476747617354487738593932700784406763128756091621633464540424...

Crossrefs

Cf. A181315, A206229, A270913, A270915, A303071, A304626, A327280.

Programs

  • Mathematica
    RealDigits[1/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Formula

Equals limit n->infinity A270913(n)^(1/n).

A301456 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x)^k)^k.

Original entry on oeis.org

1, 1, 3, 12, 49, 217, 1006, 4810, 23576, 117812, 597937, 3073874, 15972678, 83758809, 442681653, 2355678968, 12610759255, 67868269712, 366979432955, 1992755590086, 10862329206524, 59414599714958, 326009477088080, 1793977307978268, 9898072238695390, 54744525395860053, 303463833091357785
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 21 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 3*x^2 + 12*x^3 + 49*x^4 + 217*x^5 + 1006*x^6 + 4810*x^7 + 23576*x^8 + 117812*x^9 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * (1 + x^4*A(x)^4)^4 * ...
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 141*x^4/4 + 751*x^5/5 + 4064*x^6/6 + 22198*x^7/7 + 122381*x^8/8 + ... + A270922(n)*x^n/n + ...

Crossrefs

Cf. A026007, A145267, A181315, A270922, A301455.

Formula

G.f. A(x) satisfies: A(x) = exp(Sum_{k>=1} (-1)^(k+1)*x^k*A(x)^k/(k*(1 - x^k*A(x)^k)^2)).

A278428 Series reversion of g.f. (1/2)*x*(-1; -x)_inf, where (a; q)_inf is the q-Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 2, 6, 17, 46, 128, 373, 1119, 3405, 10464, 32478, 101781, 321642, 1023512, 3276326, 10543100, 34088806, 110690682, 360810160, 1180195810, 3872588051, 12743937024, 42049240694, 139082885503, 461072582522, 1531697761470, 5098246648103, 17000237006441
Offset: 1

Views

Author

Vladimir Reshetnikov, Nov 21 2016

Keywords

Comments

(1/2)*x*(-1; -x)_inf is the g.f. for A081360 shifted right.

Links

Crossrefs

Cf. A081360, A109085, A171805, A181315, A255526.

Programs

  • Mathematica
    InverseSeries[x QPochhammer[-1, -x]/2 + O[x]^35][[3]]
(* Calculation of constant c: *) 1/Sqrt[(4/s^2 - s*Derivative[0, 2][QPochhammer][-1, -s]/r) * Pi] /. FindRoot[{2*r == s*QPochhammer[-1, -s], 2*r == s^2*Derivative[0, 1][QPochhammer][-1, -s]}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

Formula

a(n) ~ c * d^n / n^(3/2), where c = 0.1211369424750398272226454930396... and d = A318204 = 3.509754327949703340437273523375193698454789733931739911... - Vaclav Kotesovec, Nov 23 2016

A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, -1, 4, 1, -17, -6, 118, -8, -876, 625, 5966, -7486, -41937, 75969, 306312, -768637, -2164992, 7487063, 14461466, -70259884, -89410774, 646971980, 459817892, -5861484630, -1128608133, 52082250637, -15894742662, -453574650852, 366848121166, 3866670213663, -5215687717614
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2018

Keywords

Examples

			G.f. A(x) = 1 - x - x^2 + 4*x^3 + x^4 - 17*x^5 - 6*x^6 + 118*x^7 - 8*x^8 - 876*x^9 + 625*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * (1 - x^4*A(x)^4)^4 * ...
log(A(x)) = -x - 3*x^2/2 + 8*x^3/3 + 13*x^4/4 - 51*x^5/5 - 120*x^6/6 + 538*x^7/7 + 781*x^8/8 - 5419*x^9/9 - 3053*x^10/10 + ... + A281267(n)*x^n/n + ...

Crossrefs

Cf. A000219, A006195, A066398, A073592, A109085, A181315, A278428, A281267, A301455, A301456, A301625.

Programs

  • Maple
    with(numtheory):
Order := 33:
Gser := solve(series(x*exp(add(sigma[2](n)*x^n/n, n = 1..32)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..32); # Peter Bala, Feb 09 2020

Formula

From Peter Bala, Feb 09 2020: (Start)
A(x) = 1/x * series reversion of ( exp( Sum_{n >= 1} sigma_2(n)*x^n/n ) ), where sigma_2(n) = A001157(n).
Equivalently, the o.g.f. A(x) satisfies [x^n](1/A(x))^n = sigma_2(n) for n >= 1. Cf. A066398. (End)
A(x) equals (1/x) * series reversion of (x * the o.g.f. for the sequence of planar partitions A000219). - Peter Bala, Feb 11 2020

A006195 Reversion of Jacobi theta_3.

Original entry on oeis.org

1, -2, 8, -40, 222, -1316, 8160, -52272, 343220, -2297682, 15623760, -107611608, 749209832, -5264005060, 37277153920, -265788870480, 1906489923022, -13747860118724, 99606357848920, -724732875917064, 5293303253527704, -38795196044205056
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000122, A109085, A181315.

Programs

  • Maple
    # Using function CompInv from A357588.
CompInv(22, n -> if n = 1 then 1 elif issqr(n-1) then 2 else 0 fi); # Peter Luschny, Oct 05 2022
  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x]^k)/(1 - x^k*A[x]^k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * (-1)^Range[0, nmax] (* Vaclav Kotesovec, Sep 27 2023 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{QPochhammer[-1, r*s] == 2*s*QPochhammer[r*s], (2* QPochhammer[r*s]*(-Log[r*s] + Log[1 - r*s] + QPolyGamma[0, 1, r*s])) / Log[r*s] + r*(Derivative[0, 1][QPochhammer][-1, r*s] - 2*s*Derivative[0, 1][QPochhammer][r*s, r*s]) == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 27 2023 *)
  • PARI
    N=66; x='x+O('x^N); /* that many terms */
Vec(serreverse(x*sum(n=-N,N,x^(n^2)))) /* show terms */ /* Joerg Arndt, May 25 2011 */

Formula

REVERT(A000122).
From Vaclav Kotesovec, Sep 27 2023: (Start)
G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + (-x)^k*A(x)^k)/(1 - (-x)^k*A(x)^k).
a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 7.86298339570590526151934790995382716105758424871057843176888470144337... and c = 0.617020565581840591336246430220953133238702598666548444780767269...
(End)

Extensions

Signs corrected by N. J. A. Sloane, Dec 24 2001

A301578 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + k*x^k*A(x)^k).

Original entry on oeis.org

1, 1, 3, 12, 48, 211, 970, 4594, 22311, 110473, 555561, 2829918, 14570666, 75708835, 396481070, 2090558864, 11089276706, 59135014252, 316836936662, 1704764660218, 9207671377450, 49904141524184, 271325301723223, 1479427708380368, 8088057338101442, 44325245804200151
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 3*x^2 + 12*x^3 + 48*x^4 + 211*x^5 + 970*x^6 + 4594*x^7 + 22311*x^8 + 110473*x^9 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + 2*x^2*A(x)^2) * (1 + 3*x^3*A(x)^3) * (1 + 4*x^4*A(x)^4) * ...
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 137*x^4/4 + 726*x^5/5 + 3896*x^6/6 + 21071*x^7/7 + 115089*x^8/8 + ... + A297322(n)*x^n/n + ...

Crossrefs

Cf. A022629, A181315, A297322, A301456, A301577.

A366026 G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + x^k*A(x)^(2*k)).

Original entry on oeis.org

1, 1, 3, 13, 64, 340, 1903, 11053, 65992, 402508, 2497207, 15709873, 99980007, 642535004, 4164018953, 27181480712, 178559253274, 1179546465168, 7830695860690, 52216823047741, 349584244515573, 2348869478981267, 15833924106623011, 107057382854642578, 725829177205070854
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 26 2023

Keywords

Crossrefs

Cf. A171802, A181315.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = Product[1 + x^k*A[x]^(2*k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
(* The constants {d,c}: *) {1/r, 1/(2*Sqrt[Pi*(1/s^2 + 2*r^2*s*Derivative[0, 2][QPochhammer][-1, r*s^2])])} /. FindRoot[{2*s == QPochhammer[-1, r*s^2], r*s*Derivative[0, 1][QPochhammer][-1, r*s^2] == 1}, {r, 1/8}, {s, 1}, WorkingPrecision -> 120]

Formula

A(x) satisfies QPochhammer(-1, x*A(x)^2) = 2*A(x).
a(n) ~ c * d^n / n^(3/2), where d = 7.2188305975020061051473056449576894316519... and c = 0.2182691546096422371919544994005940622002...

A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^k*A(x)^k)/(1 - x^k*A(x)^k))^k.

Original entry on oeis.org

1, 2, 10, 60, 398, 2820, 20892, 159868, 1253758, 10024070, 81400672, 669532924, 5566386324, 46701736772, 394910202608, 3362210548344, 28797181196766, 247955463799812, 2145088563952510, 18636002388075260, 162523319555310664, 1422259430668179592, 12485554521209720492, 109922263517662775292
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2018

Keywords

Examples

			G.f. A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 398*x^4 + 2820*x^5 + 20892*x^6 + 159868*x^7 + 1253758*x^8 + ...
G.f. A(x) satisfies: A(x) = ((1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * ...)/((1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = 2*x + 16*x^2/2 + 128*x^3/3 + 1056*x^4/4 + 8952*x^5/5 + 77200*x^6/6 + 673948*x^7/7 + 5937792*x^8/8 + ... + A270924(n)*x^n/n + ...

Crossrefs

Cf. A006195, A066398, A109085, A156616, A181315, A270924, A278428, A301455, A301456, A301624.

A301831 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 + x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, 0, 0, 6, -16, 16, -34, 217, -681, 1343, -3466, 13370, -42380, 109477, -312448, 1040248, -3267138, 9447529, -28367596, 90504001, -283611105, 861087913, -2654231074, 8386506600, -26359974392, 81902319183, -256179313766, 809890745232, -2557697524240, 8046530976599
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 27 2018

Keywords

Examples

			G.f. A(x) = 1 - x + 6*x^4 - 16*x^5 + 16*x^6 - 34*x^7 + 217*x^8 - 681*x^9 + 1343*x^10 - 3466*x^11 + ...
log(A(x)) = -x - x^2/2 - x^3/3 + 23*x^4/4 - 51*x^5/5 + 35*x^6/6 - 197*x^7/7 + ... + A281266(n)*x^n/n + ...

Crossrefs

Cf. A109085, A181315, A255528, A281266, A301455, A301456, A301624.

Formula

G.f. satisfies: A(x) = exp(Sum_{k>=1} (-1)^k*x^k*A(x)^k/(k*(1 - x^k*A(x)^k)^2)).
a(n) = [x^n] (Sum_{k>=0} A255528(k)*x^k)^(n+1)/(n + 1).
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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