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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A352640 Expansion of e.g.f. exp(3*sin(x)).

Original entry on oeis.org

1, 3, 9, 24, 45, -24, -747, -3864, -7623, 48576, 548001, 2175360, -5269275, -133496448, -785549331, 789324288, 52523738865, 398157422592, -157735851975, -31426593116160, -278010988509411, 108354846277632, 27060485795905221, 257882404940021760
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Cf. A002017, A136630, A352639.
Cf. A352280, A352638.

Programs

  • Mathematica
    With[{m = 23}, Range[0, m]! * CoefficientList[Series[Exp[3*Sin[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*sin(x))))
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*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 * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^k * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025

A381145 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-sin(x)) ).

Original entry on oeis.org

1, 1, 3, 15, 105, 937, 10059, 124607, 1720593, 25578001, 391041299, 5628440015, 55397475705, -847789025159, -93469767131685, -5040670692970753, -236210967512228575, -10629917015586704351, -475183316832486106589, -21394016956935371375601, -975459739630268065696887
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Links

Crossrefs

Cf. A002017, A381142, A381148.
Cf. A136630, A185690, A334856.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp( sin(x * A(x)) ).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A381148 E.g.f. A(x) satisfies A(x) = exp( sin(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 0, -11, -88, -459, -560, 27945, 502336, 5223945, 18968576, -671465123, -20909349888, -345616002627, -2437013715968, 65881260463697, 3252353828442112, 76987773739473809, 873339053357432832, -17521718791602049595, -1354633521318944473088
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Crossrefs

Cf. A002017, A381142, A381145.
Cf. A136630, A185690.

Programs

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

Formula

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A009209 Expansion of e.g.f.: exp(sin(x))/exp(x).

Original entry on oeis.org

1, 0, 0, -1, 0, 1, 10, -1, -56, -279, 246, 4619, 14388, -53195, -556478, -864865, 13276912, 90192753, -72903378, -3987888493, -16957067028, 101506932205, 1411655530330, 2206092853799, -70455418153496, -549095655588183, 1428569363164230
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A002017, A009227.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Sin[x]]/Exp[x],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jun 27 2012 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(sin(x))/exp(x))) \\ Michel Marcus, Apr 09 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * a(n-2*k-1). - Ilya Gutkovskiy, Apr 09 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Definition clarified by Harvey P. Dale, Jun 27 2012

A352377 Expansion of e.g.f. exp(1 - cos(x) + sin(x)).

Original entry on oeis.org

1, 1, 2, 3, 5, 2, -17, -105, -302, -323, 2735, 21318, 74513, 5345, -1876118, -13036317, -35542499, 183591298, 2771934527, 14515620855, -4104116566, -739297426531, -6244977674825, -14587702161978, 240078040966369, 3207383844181633, 14652985540658834, -87474514259307453, -2013684557381588299
Offset: 0

Views

Author

Seiichi Manyama, Mar 14 2022

Keywords

Links

Crossrefs

Cf. A002017, A260884.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(1-cos(x)+sin(x))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, (-1)^((k-1)\2)*binomial(n-1, k-1)*a(n-k)));

Formula

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

A381278 Expansion of e.g.f. exp(sin(3*x) / 3).

Original entry on oeis.org

1, 1, 1, -8, -35, -8, 1117, 6328, -19943, -513728, -2096711, 30574720, 447401845, 23791744, -59033858219, -527680180736, 4971322421425, 144677554315264, 430091284739185, -27641200139694080, -398305237630617971, 2876369985206861824, 145441158283475935309
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Cf. A002017, A009210.
Cf. A136630, A352640, A381286.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (3*I)^(n-k)*a136630(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-9)^k * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} (3*i)^(n-k) * A136630(n,k), where i is the imaginary unit.

A007301 From expansion of exp(sin x).

Original entry on oeis.org

1, 1, -3, -3, 217, -2951, 5973, 1237173, -52635599, 1126610929, 20058390573, -3920482183827, 256734635981833, -8529964147714967, -383670903748980603, 99786978811973617701, -10380441423510120242591, 594094494902602207843297
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Programs

  • Mathematica
    With[{nn=40},Take[CoefficientList[Series[Exp[Sin[x]],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Dec 22 2018 *)

Formula

a(n) = A002017(2n).

Extensions

Extended with signs by Christian G. Bower, Oct 1998

A009282 Expansion of e.g.f.: exp(x + sin(x)).

Original entry on oeis.org

1, 2, 4, 7, 8, -7, -74, -209, -40, 2441, 10954, 6667, -185524, -1041739, -510530, 26716831, 158378768, -47260303, -6226501934, -34323431789, 71521515284, 2087815263725, 9437342779558, -55046954388953, -923175950185960, -2736047549718871, 43387645616657338
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A002017, A009209, A009283.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x+Sin[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 23 2014 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x + sin(x)))) /* Joerg Arndt, Sep 01 2012 */

Formula

E.g.f.: exp(x + sin(x)).
a(0) = 1; a(n) = a(n-1) + Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * a(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Name corrected by Arkadiusz Wesolowski, Sep 01 2012

A352145 Expansion of e.g.f. exp(-1 + cos(x) + sin(x)).

Original entry on oeis.org

1, 1, 0, -3, -5, 12, 71, 7, -1028, -2573, 14793, 100188, -128831, -3445791, -5741800, 113954461, 601512787, -3296210612, -41316895641, 37322755431, 2570678600548, 6983413204755, -149303353515823, -1080122148248420, 7405149869523649, 119115584584019713
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2022

Keywords

Links

Crossrefs

Cf. A002017, A005046, A292935, A352377.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-1+cos(x)+sin(x))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, (-1)^(k\2)*binomial(n-1, k-1)*a(n-k)));

Formula

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

A352639 Expansion of e.g.f. exp(2*sin(x)).

Original entry on oeis.org

1, 2, 4, 6, 0, -46, -192, -266, 1792, 14114, 34816, -171930, -2027520, -6522382, 34750464, 496296022, 1748500480, -12731696062, -186550845440, -617309234490, 7292215885824, 99199654760978, 248883934396416, -5836506132182090, -69729013345550336
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Cf. A002017, A136630, A352640.
Cf. A007289, A352279, A352642.

Programs

  • Mathematica
    With[{m = 24}, Range[0, m]! * CoefficientList[Series[Exp[2*Sin[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sin(x))))
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*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 * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^k * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025
Previous Showing 11-20 of 24 results. Next

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

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