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.

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A385311 Expansion of e.g.f. 1/(1 - 3 * x * cos(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 25, 232, 2805, 41920, 744933, 15340416, 359136073, 9419223040, 273558859409, 8714789788672, 302151400126589, 11326084055150592, 456421403198919325, 19677025400034590720, 903660903945306053137, 44042354270955276599296, 2270411632567521580120713
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385310.
Cf. A380017, A385305, A385307, A385309.
Cf. A007559, A185951, A352647, A385284.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

A352642 Expansion of e.g.f. exp(2 * x * cos(x)).

Original entry on oeis.org

1, 2, 4, 2, -32, -198, -416, 2634, 30720, 107378, -605696, -10282094, -46020608, 304968874, 6121832448, 29994597338, -279697555456, -5729595393310, -26849178681344, 401845799334690, 7714801999937536, 29062583111892506, -812705956979802112
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Links

Crossrefs

Cf. A009189, A185951, A352643.

Programs

  • Mathematica
    With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[2*x*Cos[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*x*cos(x))))
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*binomial(n-1, 2*k)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^k * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025

A352643 Expansion of e.g.f. exp(3 * x * cos(x)).

Original entry on oeis.org

1, 3, 9, 18, -27, -552, -3051, -3504, 102825, 1043712, 3192129, -41548416, -653192883, -3033406464, 31367500173, 670266381312, 3916411302609, -40460110970880, -1038593550985479, -6810646726410240, 82445831323038261, 2280185182260854784, 15300402721484153733
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Links

Crossrefs

Cf. A009189, A185951, A352642.

Programs

  • Mathematica
    With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[3*x*Cos[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*x*cos(x))))
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*binomial(n-1, 2*k)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^k * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025

A381175 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cos(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 69, 1224, 29465, 898320, 33187133, 1441200768, 71956238769, 4061414246400, 255737764687669, 17773804761259008, 1351494159065894857, 111608708333568036864, 9947544079380663728685, 951770403836914402099200, 97301151510219112917218657, 10585077723403580668983902208
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A381173, A381174, A381176.
Cf. A364984, A381172.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+2*k+1, k)/(n+2*k+1)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381209 Expansion of e.g.f. 1/(1 - x*cos(x))^3.

Original entry on oeis.org

1, 3, 12, 51, 216, 735, 0, -39081, -575232, -6047973, -48314880, -189159333, 3046957056, 99745485879, 1789140627456, 23433663134655, 185580069027840, -1250544374605389, -94781673979379712, -2543434372808424957, -47763303489939701760, -586864592847636893937
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A377530, A381207, A381210, A381211.
Cf. A352252, A381208.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+2)!*I^(n-k)*a185951(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381280 Expansion of e.g.f. 1/(1 - x * cosh(2*x)).

Original entry on oeis.org

1, 1, 2, 18, 120, 920, 10320, 126448, 1714048, 27073152, 472354560, 8989147904, 187690331136, 4245706716160, 103239264593920, 2691918892861440, 74885151106498560, 2212607133043884032, 69227613551324233728, 2286465386258267176960, 79487593489348266557440
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A381281.
Cf. A185951, A201939.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*2^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 2^(n-k) * A185951(n,k).
a(n) ~ sqrt(Pi) * 2^(n + 5/2) * n^(n + 1/2) / ((1 + sinh(r))^2 * exp(n) * r^(n+2)), where r = A201939. - Vaclav Kotesovec, Apr 19 2025

A381283 Expansion of e.g.f. 1/(1 - x * cos(3*x)).

Original entry on oeis.org

1, 1, 2, -21, -192, -1095, 7200, 243747, 3088512, 1360881, -874437120, -21701765349, -186175604736, 5870711879721, 292185085151232, 5507319584787795, -38951106749890560, -6402114772676575263, -212680600451474522112, -1602903494245708491957, 197042528380347210792960
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A352252, A381282.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*(3*I)^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-9)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * (3*i)^(n-k) * A185951(n,k), where i is the imaginary unit.

A381449 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^2 ).

Original entry on oeis.org

1, 2, 10, 90, 1224, 22450, 517920, 14395514, 468414464, 17474840226, 735559614720, 34491849224602, 1783268816102400, 100786369113730898, 6182264844496971776, 409065938149354422330, 29043282491002728284160, 2202461172795524123296834, 177675452451923238962528256
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A381171, A381450.
Cf. A185951, A381447.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+2, k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cosh(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381447.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * A185951(n,k).

A385281 Expansion of e.g.f. 1/(1 - 2 * x * cosh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 27, 249, 2825, 41355, 708883, 13888497, 309267729, 7698772755, 211585744139, 6367841422569, 208299923870233, 7357493992966299, 279095125351544835, 11316313498670411745, 488403056864943302177, 22355228989851909617187, 1081663315375339026249211
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385282.
Cf. A001586, A235134, A380155, A385283.
Cf. A001147, A185951.
Cf. A069814.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*2^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A185951(n,k), where A185951(n,0) = 0^n.
a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + r*sqrt(1 - r^2)) * exp(n) * r^n), where r = A069814. - Vaclav Kotesovec, Jun 24 2025

A385308 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 18, 141, 1400, 17055, 245392, 4070073, 76483584, 1606033755, 37267953536, 947051118981, 26156846230528, 780174007426359, 24992424003517440, 855795857724702705, 31193844533488074752, 1205893835653392258867, 49280187764171870470144, 2122704756621224015194365
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385309.
Cf. A380015, A385304, A385306, A385310.
Cf. A001147, A185951, A352648, A385281.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * A185951(n,k), where A185951(n,0) = 0^n.
a(n) ~ sqrt(2) * n^n / (sqrt(1 + r*sqrt(1 - 4*r^2)) * exp(n) * r^n), where r = 0.452787214835453627588998503316635625709288535855... is the root of the equation 2*r*cosh(r) = 1. - Vaclav Kotesovec, Jun 28 2025
Previous Showing 11-20 of 77 results. Next

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