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 12 results. Next

A380017 Expansion of e.g.f. 1/(1 - 3*x*exp(x))^(1/3).

Original entry on oeis.org

1, 1, 6, 55, 716, 12085, 250726, 6172915, 175903400, 5694587209, 206438732810, 8284550317351, 364605758728828, 17461047965591581, 903964982917764782, 50306323769422679995, 2994799872257498255696, 189906103853462927405329, 12779300537432602189228306
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A006153, A380015, A380019.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-3)^k*k^(n-k)*binomial(-1/3, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (-3)^k * k^(n-k) * binomial(-1/3,k)/(n-k)!.
a(n) ~ sqrt(2*Pi) * n^(n - 1/6) / (Gamma(1/3) * (1 + LambertW(1/3))^(1/3) * exp(n) * LambertW(1/3)^n). - Vaclav Kotesovec, Jan 23 2025

A385310 Expansion of e.g.f. 1/(1 - 2 * x * cos(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 12, 69, 500, 4455, 46928, 571977, 7914384, 122585355, 2100940864, 39470867469, 806555184448, 17808628411119, 422498774818560, 10717948285126545, 289501146405400832, 8295124400250875667, 251300745071590317056, 8025654235707259740885, 269482309052945201181696
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385311.
Cf. A380015, A385304, A385306, A385308.
Cf. A001147, A185951, A352646, A385283.

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)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

A380155 Expansion of e.g.f. 1/sqrt(1 - 2*x*exp(2*x)).

Original entry on oeis.org

1, 1, 7, 63, 785, 12545, 244407, 5619775, 148977313, 4473497601, 150078670055, 5563415292479, 225832882678449, 9962766560986369, 474619650950131351, 24283168467229957695, 1327993894505461755713, 77305844496338607597569, 4772660185400974888323015
Offset: 0

Views

Author

Seiichi Manyama, Jan 13 2025

Keywords

Crossrefs

Cf. A006153, A380156.
Cf. A380014, A380015.

Programs

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

Formula

a(n) = 2^n * n! * Sum_{k=0..n} (-1)^k * k^(n-k) * binomial(-1/2,k)/(n-k)!.
a(n) == 1 (mod 2).
a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + LambertW(1)) * exp(n) * LambertW(1)^n). - Vaclav Kotesovec, Jan 23 2025

A385304 Expansion of e.g.f. 1/(1 - 2 * sinh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 16, 117, 1096, 12543, 169576, 2644617, 46735936, 922993083, 20145579136, 481555537917, 12511452674176, 351058439096823, 10579734482269696, 340820224678288017, 11687491783287586816, 425075150516293691763, 16343274366458168160256, 662325275389743380902917
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A006154, A385305.
Cf. A380015, A385306, A385308, A385310.
Cf. A001147, A136630, A235134, A364822.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * A136630(n,k).
a(n) ~ sqrt(2) * n^n / (5^(1/4) * exp(n) * log((1 + sqrt(5))/2)^(n + 1/2)). - Vaclav Kotesovec, Jun 28 2025

A385306 Expansion of e.g.f. 1/(1 - 2 * sin(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 14, 93, 796, 8343, 103424, 1479993, 24008656, 435364683, 8726775584, 191601310293, 4572794295616, 117871476051423, 3263515787807744, 96591500816346993, 3043368045293138176, 101702692426476460563, 3592948632452749243904, 133794496537591022166093
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A000111, A385307.
Cf. A380015, A385304, A385308, A385310.
Cf. A001147, A001586, A007289, A136630.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/Sqrt[1-2Sin[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 09 2025 *)
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
a(n) ~ 2^(n+1) * 3^(n + 1/4) * n^n / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Jun 28 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

A380019 Expansion of e.g.f. 1/(1 - 4*x*exp(x))^(1/4).

Original entry on oeis.org

1, 1, 7, 78, 1249, 26100, 673101, 20655082, 735030913, 29759100264, 1350726180085, 67929497104326, 3749296817347137, 225321905599163308, 14646040616615433949, 1023818460912628352490, 76589660469522857865601, 6105092923000191785913552, 516586509938516858800548453
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A006153, A380015, A380017.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-4)^k*k^(n-k)*binomial(-1/4, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (-4)^k * k^(n-k) * binomial(-1/4,k)/(n-k)!.

A380035 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*exp(x*A(x)) ).

Original entry on oeis.org

1, 1, 5, 42, 517, 8420, 171201, 4181128, 119339081, 3900501648, 143703797725, 5893732487456, 266358266633229, 13153210420876864, 704697559381904921, 40714369264722337920, 2523456287242464370321, 167019778198736205721856, 11757749450929277192860725
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Cf. A380015, A380042.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} 2^k * k^(n-k) * binomial(n/2+k/2+1/2,k)/( (n+k+1)*(n-k)! ).

A380042 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*exp(x*A(x)^2) ).

Original entry on oeis.org

1, 1, 5, 48, 697, 13640, 336771, 10053778, 352334753, 14183529480, 645073504435, 32715111226886, 1830671281889649, 112049330303532388, 7446824171300128811, 534068807341887943770, 41111698162393482004801, 3381089519620006418116976, 295869084136630532211207843
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Cf. A380015, A380035.
Cf. A161633, A380043.
Cf. A201470.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(serreverse(x/(1+2*x*exp(x)))/x)))
  • PARI
    a(n) = n!*sum(k=0, n, 2^k*k^(n-k)*binomial(n+1/2, k)/(n-k)!)/(2*n+1);

Formula

E.g.f.: sqrt( (1/x) * Series_Reversion(x/(1 + 2*x*exp(x))) ).
a(n) = (n!/(2*n+1)) * Sum_{k=0..n} 2^k * k^(n-k) * binomial(n+1/2,k)/(n-k)!.

A380133 Expansion of e.g.f. sqrt(1 + 2*x*exp(x)).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, -9, 70, -335, 504, 11935, -182094, 1525833, -4911764, -99495473, 2430329070, -29988416159, 158542630224, 2868272912511, -102775471991126, 1714422613948345, -13166449628575404, -209400601689898289, 10598981162761786950, -227206614609529433199
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A028310, A380134.
Cf. A380050, A380093.
Cf. A380015.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} 2^k * k^(n-k) * binomial(1/2,k)/(n-k)!.
Showing 1-10 of 12 results. Next

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

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