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 1-6 of 6 results.

A352117 Expansion of e.g.f. 1/sqrt(2 - exp(2*x)).

Original entry on oeis.org

1, 1, 5, 37, 377, 4921, 78365, 1473277, 31938737, 784384561, 21523937525, 652667322517, 21672312694697, 782133969325801, 30481907097849485, 1275870745561131757, 57083444567425884257, 2718602143583362124641, 137315150097164841942245
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2022

Keywords

Crossrefs

Cf. A000670, A097397, A216794, A352118, A352119, A305404, A346982 - A346985.

Programs

  • Mathematica
    m = 18; Range[0, m]! * CoefficientList[Series[(2 - Exp[2*x])^(-1/2), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
  • Maxima
    a[n]:=if n=0 then 1 else sum(a[n-k]*(1-k/n/2)*binomial(n,k)*2^k,k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Sep 06 2023 */
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(2-exp(2*x))))
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*prod(j=0, k-1, 2*j+1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * Stirling2(n,k).
a(n) ~ 2^n * n^n / (log(2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Mar 05 2022
Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x/(1 - 4*x/(1 - 3*x/(1 - 8*x/(1 - ... - (2*n-1)*x/(1 - 4*n*x/(1 - ... ))))))). Cf. A346982. - Peter Bala, Aug 22 2023
For n > 0, a(n) = Sum_{k=1..n} a(n-k)*(1-k/n/2)*binomial(n,k)*2^k. - Tani Akinari, Sep 06 2023
a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-2)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 18 2023

A352070 Expansion of e.g.f. 1/(1 - log(1 + 3*x))^(1/3).

Original entry on oeis.org

1, 1, 1, 10, 10, 604, -1844, 107344, -1201400, 42193576, -875584376, 29853569008, -880141783184, 32865860907424, -1216481572723616, 51296026356128512, -2244334822166729600, 106984479644794783360, -5358207684820194270080, 286466413246622566048000
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2022

Keywords

Crossrefs

Cf. A006252, A097397, A352073.
Cf. A007559, A133480, A352113, A352118.

Programs

  • Mathematica
    m = 19; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 3*x])^(-1/3), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
Table[Sum[3^(n-k) * Product[3*j+1, {j,0,k-1}] * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 07 2023 *)
  • Maxima
    a[n]:=if n=0 then 1 else n!*sum(a[n-k]*(2/n/3-1/k)*(-3)^k/(n-k)!,k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Sep 07 2023 */
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+3*x))^(1/3)))
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*prod(j=0, k-1, 3*j+1)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} 3^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling1(n,k).
For n > 0, a(n) = n!*Sum_{k=1..n} a(n-k)*(2/n/3-1/k)*(-3)^k/(n-k)!. - Tani Akinari, Sep 07 2023
a(n) ~ -(-1)^n * 3^(n-1) * n! / (n * log(n)^(4/3)) * (1 - 4*(1+gamma)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 07 2023

A352073 Expansion of e.g.f. 1/(1 - log(1 + 4*x))^(1/4).

Original entry on oeis.org

1, 1, 1, 17, 1, 1889, -12415, 631665, -11224575, 461864385, -13754112255, 596055636945, -24148300842495, 1181210529292065, -59009709972278655, 3297137505670374705, -193318225258785780735, 12263541239089421903745, -820804950905249837195775
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2022

Keywords

Crossrefs

Cf. A006252, A097397, A352070.
Cf. A007696, A008545, A352119.

Programs

  • Mathematica
    m = 18; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 4*x])^(-1/4), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+4*x))^(1/4)))
  • PARI
    a(n) = sum(k=0, n, 4^(n-k)*prod(j=0, k-1, 4*j+1)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-4)^k * (3/4 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 18 2023

A352075 Expansion of e.g.f. sqrt(1 - log(1 - 2*x)).

Original entry on oeis.org

1, 1, 1, 5, 25, 209, 1961, 23589, 321105, 5100801, 90384369, 1792247973, 39011436201, 928869511569, 23953711043289, 666047439187077, 19847286284835105, 631267636613496705, 21339849019758468705, 764149215124570567365, 28891697037933017586105
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2022

Keywords

Crossrefs

Cf. A352113, A352114.
Cf. A097397.

Programs

  • Mathematica
    m = 20; Range[0, m]! * CoefficientList[Series[(1 - Log[1 - 2*x])^(1/2), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(1-log(1-2*x))))
  • PARI
    a(n) = sum(k=0, n, (-2)^(n-k)*prod(j=0, k-1, -2*j+1)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (-2*j+1)) * Stirling1(n,k).
a(n) ~ n! * 2^(n-1) / (sqrt(log(n)) * n) * (1 - (gamma + 1)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 05 2022

A095398 Number of steps required to reach 1 for iterated modified juggler map of A095396.

Original entry on oeis.org

0, 1, 7, 2, 6, 8, 10, 3, 7, 3, 5, 7, 9, 7, 9, 9, 11, 9, 11, 11, 13, 11, 13, 4, 10, 4, 6, 8, 10, 8, 10, 4, 8, 4, 8, 4, 12, 6, 12, 6, 8, 8, 12, 8, 12, 8, 10, 10, 14, 10, 12, 10, 14, 8, 12, 8, 10, 8, 14, 10, 12, 10, 12, 10, 12, 10, 12, 10, 14, 10, 12, 12, 16, 12, 18, 12, 18, 10, 12, 10, 16
Offset: 1

Views

Author

Labos Elemer, Jun 18 2004

Keywords

Comments

Parallel to A007320.

Examples

			n=37: the trajectory is {37, 225, 3375, 196069, 86818724, 196068, 3374, 224, 36, 10, 4, 2, 1}, number of required steps is a[37]=13-1=12.

Crossrefs

Cf. A007320, A094683, A094716, A095396, A097397.

Programs

  • Mathematica
    d[x_]:=d[x]=(1-Mod[x, 2])*Floor[N[x^(2/3), 50]] +Mod[x, 2]*Floor[N[x^(3/2), 50]];d[1]=1; fd[x_]:=Delete[FixedPointList[d, x], -1] Table[Max[fd[w]], {w, 1, m}]
Table[Length[NestWhileList[If[EvenQ[#],Floor[#^(2/3)],Floor[#^(3/2)]]&, n, #!=1&]]-1,{n,90}] (* Harvey P. Dale, Dec 28 2018 *)

A367423 Expansion of e.g.f. 1 / sqrt(1 + log(1 - 2*x)).

Original entry on oeis.org

1, 1, 5, 41, 465, 6729, 118437, 2455809, 58630401, 1584058161, 47783202213, 1591924168185, 58055219617425, 2300356943749305, 98409722434170885, 4520749198158270225, 221954573405993807745, 11598560660172502840545, 642753897983638032821445
Offset: 0

Views

Author

Seiichi Manyama, Nov 18 2023

Keywords

Crossrefs

Cf. A097397, A352117.

Programs

  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*prod(j=0, k-1, 2*j+1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * |Stirling1(n,k)|.
a(0) = 1; a(n) = Sum_{k=1..n} 2^k * (1 - 1/2 * k/n) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ 2^(n + 1/2) * n^n / (exp(1) - 1)^(n + 1/2). - Vaclav Kotesovec, Jun 09 2025
Showing 1-6 of 6 results.

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