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.

A369557 Expansion of Sum_{n>=0} Product_{k=0..n} (x^k + x^(n-k)).

Original entry on oeis.org

3, 4, 2, 6, 3, 4, 9, 4, 8, 6, 13, 8, 12, 12, 10, 22, 13, 22, 14, 26, 20, 34, 23, 32, 36, 34, 42, 36, 59, 38, 67, 46, 75, 56, 82, 66, 98, 84, 100, 102, 105, 126, 116, 152, 119, 184, 136, 202, 154, 230, 181, 256, 203, 276, 250, 306, 285, 326, 342, 348, 398, 374, 463, 404, 525, 438, 610, 486, 666, 542, 744, 610
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Examples

			G.f.: A(x) = 3 + 4*x + 2*x^2 + 6*x^3 + 3*x^4 + 4*x^5 + 9*x^6 + 4*x^7 + 8*x^8 + 6*x^9 + 13*x^10 + 8*x^11 + 12*x^12 + ...
where
A(x) = (1 + 1) + (1 + x)*(x + 1) + (1 + x^2)*(x + x)*(x^2 + 1) + (1 + x^3)*(x + x^2)*(x^2 + x)*(x^3 + 1) + (1 + x^4)*(x + x^3)*(x^2 + x^2)*(x^3 + x)*(x^4 + 1) + (1 + x^5)*(x + x^4)*(x^2 + x^3)*(x^3 + x^2)*(x^4 + x)*(x^5 + 1) + ...
Also,
A(1/x) = (1 + 1) + (1 + x)*(x + 1)/x^2 + (1 + x^2)*(x + x)*(x^2 + 1)/x^6 + (1 + x^3)*(x + x^2)*(x^2 + x)*(x^3 + 1)/x^12 + (1 + x^4)*(x + x^3)*(x^2 + x^2)*(x^3 + x)*(x^4 + 1)/x^20 + (1 + x^5)*(x + x^4)*(x^2 + x^3)*(x^3 + x^2)*(x^4 + x)*(x^5 + 1)/x^30 + ...
For example, at x = 1/2,
A(1/2) = 2 + 9/2^2 + 100/2^6 + 2916/2^12 + 231200/2^20 + 50808384/2^30 + 31258240000/2^42 + 54112148361216/2^56 + 264265663201280000/2^72 + ... + A369673(n)/2^(n*(n+1)) + ... = 6.80013983505192354264...
SPECIFIC VALUES.
A(t) = 4 at t = 0.21135479438007733067820905390237206358880...
A(t) = 5 at t = 0.35111207737762337157349938790010474080253...
A(t) = 6 at t = 0.44509902476179757380223857309576063477813...
A(3/4) = 18.04139246037655138841324835985762487898724341...
A(2/3) = 11.59103511448176661974748662249737201844158309...
A(Phi) = 9.595623356758087506923478384122062088751068609...
A(1/2) = 6.800139835051923542641455169580774467247971025...
A(1/3) = 4.847274134844057155467506697748724715389597193...
A(1/4) = 4.236976626306045459467696438142250301516563681...
A(1/5) = 3.934732308501055907377639201049737298238369356...

Links

Crossrefs

Cf. A369673, A369674, A369675, A369676, A376542, A376621.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[Product[x^j + x^(k - j), {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 08 2024 *)
nmax = 100; CoefficientList[Series[-1 + 2*Sum[x^(k^2) * Product[1 + x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}] + Sum[x^((k-1)*k) * Product[1 + x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 27 2024 *)
  • PARI
    {a(n) = my(A = sum(m=0,n+1, prod(k=0,m, x^k + x^(m-k)) +x*O(x^n) )); polcoeff(A,n)}
for(n=0,70, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} Product_{k=0..n} (x^k + x^(n-k)).
(2) A(x) = Sum_{n>=0} x^(n*(n+1)) * Product_{k=0..n} (1/x^k + 1/x^(n-k)).
(3) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * Product_{k=0..n} (1 + x^(n-2*k)).
From Vaclav Kotesovec, Sep 29 2024: (Start)
a(n) ~ c * d^sqrt(n) / sqrt(n), where d = A376621 = 2.7510850908889199... and c = sqrt((1 + ((197 - sqrt(27/31)) / 62)^(1/3) + ((197 + sqrt(27/31)) / 62)^(1/3))/3) = 1.146046709280363...
a(n) ~ 4*A376542(n). (End)

A333198 Decimal expansion of a constant related to the asymptotics of A306734 and A333179.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 11 2020

Keywords

Examples

			1.86429525435844065924747599856112246877295244507368421574403...

Crossrefs

Cf. A306734, A333179, A333180, A333181, A357471, A376621.

Programs

  • Mathematica
    RealDigits[E^Sqrt[4*Log[r]^2/3 + 4*PolyLog[2, 1-r] - Pi^2/3] /. r -> (2 - 5*(2/(-11 + 3*Sqrt[69]))^(1/3) + ((-11 + 3*Sqrt[69])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A306734(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A333179(n)^(1/sqrt(n)).
Equals exp(sqrt(4*log(r)^2/3 + 4*polylog(2, 1-r) - Pi^2/3)), where r = 1 - A357471 = 0.4301597090019467340886... is the real root of the equation r^2 = (1-r)^3. - Vaclav Kotesovec, Oct 07 2024

Extensions

More digits from Vaclav Kotesovec, Oct 07 2024

A376658 Decimal expansion of a constant related to the asymptotics of A376624 and A376625.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 01 2024

Keywords

Examples

			8.46018724425296480975230009888917594335470635951014367622821158904321498...

Crossrefs

Cf. A072223, A333198, A376621, A376624, A376625, A376659, A376660, A377075.
Cf. A356032.

Programs

  • Mathematica
    RealDigits[E^(Sqrt[2*Log[r]^2 + 4*PolyLog[2, Sqrt[r]]]) /. r -> 1/(2*Sqrt[3/(4 + ((155 - 3*Sqrt[849])/2)^(1/3) + ((155 + 3*Sqrt[849])/2)^(1/3))]) - Sqrt[8/3 - ((155 - 3*Sqrt[849])/2)^(1/3)/3 - ((155 + 3*Sqrt[849])/2)^(1/3)/3 + 2*Sqrt[3/(4 + ((155 - 3*Sqrt[849])/2)^(1/3) + ((155 + 3*Sqrt[849])/2)^(1/3))]]/2, 10, 105][[1]]

Formula

Equals exp(sqrt(2*(log(r)^2 + 2*polylog(2, sqrt(r))))), where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.
Equals limit_{n->infinity} A376624(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376625(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A377075(n)^(1/sqrt(n)).
Equals exp(2*sqrt(2*log(A356032)^2 + polylog(2, A356032))).

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 3, 3, 3, 4, 4, 5, 5, 7, 9, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 19, 24, 23, 25, 27, 28, 31, 32, 33, 37, 40, 42, 44, 47, 52, 54, 59, 62, 67, 75, 75, 80, 87, 90, 95, 102, 109, 114, 119, 127, 134, 142, 150, 159, 171, 178, 187, 199, 211
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 29 2024

Keywords

Links

Crossrefs

Cf. A053258, A216222, A306734, A369557, A376542, A376581, A376621.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.390989767113799449629...
a(n) ~ c * A376542(n), where c = (108 + 12*sqrt(93))^(1/3)/3 - 4/(108 + 12*sqrt(93))^(1/3) = 1.364655607... is the real root of the equation c*(4 + c^2) = 8.
a(n) ~ c * A369557(n), where c = A347178 = -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3) / sqrt(3) = 0.3411639019... is the real root of the equation 2*c*(1 + 4*c^2) = 1.
a(n) ~ A376631(n) * (A376621/A376660)^sqrt(n).

A376660 Decimal expansion of a constant related to the asymptotics of A376630 and A376631.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 01 2024

Keywords

Examples

			2.045390691852050639893704244342601252265948793467833187994662870934455617...

Crossrefs

Cf. A333198, A376621, A376630, A376631, A376658, A376659, A376815.
Cf. A088559.

Programs

  • Mathematica
    RealDigits[E^Sqrt[3*Log[r]^2/4 + 2*PolyLog[2, r^(1/2)] - Pi^2/6] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A376630(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376631(n)^(1/sqrt(n)).
Equals A376815^(1/2). - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2/4 + 2*polylog(2, r^(1/2)) - Pi^2/6)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A376659 Decimal expansion of a constant related to the asymptotics of A376626 and A376627.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 01 2024

Keywords

Examples

			3.33526020703708086029122448156335246730884987099277968...

Crossrefs

Cf. A075778, A333198, A376621, A376626, A376627, A376658, A376660.

Programs

  • Mathematica
    RealDigits[E^Sqrt[6*Log[r]^2 + 2*PolyLog[2, 1 - r^2]] /. r -> (-1 + ((25 - 3*Sqrt[69])/2)^(1/3) + ((25 + 3*Sqrt[69])/2)^(1/3))/3, 10, 105][[1]]

Formula

Equals exp(sqrt(2*(3*log(r)^2 + polylog(2, 1 - r^2)))), where r = A075778 = 0.7548776662466927600495088963585286918946... is the real root of the equation r^2*(1+r) = 1.
Equals limit_{n->infinity} A376626(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376627(n)^(1/sqrt(n)).

A376631 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j)).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 2, 0, 3, 0, 2, 1, 3, 1, 3, 1, 2, 3, 2, 3, 2, 4, 1, 5, 2, 5, 2, 6, 1, 7, 2, 7, 3, 6, 4, 7, 5, 6, 7, 6, 7, 7, 9, 5, 11, 5, 12, 6, 14, 5, 15, 6, 16, 7, 17, 7, 18, 9, 18, 11, 19, 12, 20, 14, 19, 17, 19, 19, 20, 23, 18, 27, 18, 29, 20, 32, 19
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A000009, A053258, A053261, A264905, A333179, A376580, A376632, A376660.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j)).
a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.39098976711379944962936707496887239986756106886318...
a(n) ~ A376580(n) * (A376660/A376621)^sqrt(n).

A376623 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^(2*j-1))^2.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 4, 2, 2, 4, 2, 2, 4, 4, 5, 4, 6, 10, 6, 6, 9, 8, 9, 8, 10, 12, 11, 14, 14, 16, 17, 18, 23, 20, 22, 26, 26, 26, 27, 34, 31, 32, 39, 40, 43, 42, 48, 54, 55, 56, 63, 72, 68, 74, 80, 84, 88, 90, 101, 104, 109, 112, 121, 130, 132, 144, 152, 160
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A376542, A376580, A376621, A376626, A376629.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1))*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*(1 + x^(2*k - 1))*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j-1))^2.
a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = sqrt(sinh(arcsinh(3*sqrt(93)/2)/3)) / (sqrt(2)*93^(1/4)) = 0.26678318911398751342...

A376815 Decimal expansion of a constant related to the asymptotics of A376812.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 05 2024

Keywords

Examples

			4.18362308231501037592434207471436289895638697707035887857832710...

Crossrefs

Cf. A333198, A376621, A376658, A376659, A376660, A376812.
Cf. A088559.

Programs

  • Mathematica
    RealDigits[E^Sqrt[3*Log[r]^2 + 8*PolyLog[2, r^(1/2)] - 2*Pi^2/3] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A376812(n)^(1/sqrt(n)).
Equals A376660^2. - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2 + 8*polylog(2, r^(1/2)) - 2*Pi^2/3)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A376152 Decimal expansion of a constant related to the asymptotics of A376530.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 09 2024

Keywords

Examples

			4.988020876600903805335224460790773050832038156091687962387444991919...

Crossrefs

Cf. A376530, A376621.

Programs

  • Mathematica
    RealDigits[E^(2*Sqrt[Log[r]^2 + 2*PolyLog[2, 1-r] - 2*PolyLog[2, 1-r^3]/3]) /. r -> (-1 - 2/(17 + 3*Sqrt[33])^(1/3) + (17 + 3*Sqrt[33])^(1/3))/3, 10, 120][[1]]

Formula

Equals limit_{n->infinity} A376530(n)^(1/sqrt(n)).
Equals exp(2*sqrt(log(r)^2 + 2*polylog(2, 1-r) - 2*polylog(2, 1-r^3)/3)), where r = A192918 = 0.54368901269207636157085597180174... is the real root of the equation r^2 * (1-r^3)^2 = (1-r)^2.
Showing 1-10 of 10 results.

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