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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A385305 Expansion of e.g.f. 1/(1 - 3 * sinh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 29, 296, 3921, 63904, 1236509, 27700096, 705098241, 20100847104, 634406699389, 21959759364096, 827184049670161, 33684401687855104, 1474548883501060669, 69051807696652599296, 3444499079760040247681, 182339939994632235515904, 10209271857672376613472349
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A006154, A385304.
Cf. A380017, A385307, A385309, A385311.
Cf. A007559, A136630, A235135.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * A136630(n,k).
a(n) ~ sqrt(Pi) * 2^(1/3) * n^(n - 1/6) / (5^(1/6) * Gamma(1/3) * exp(n) * log((1 + sqrt(10))/3)^(n + 1/3)). - 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

A385307 Expansion of e.g.f. 1/(1 - 3 * sin(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 27, 264, 3361, 52704, 981707, 21176704, 519150241, 14255163904, 433384277787, 14451212550144, 524406240059521, 20572970822959104, 867641565719168267, 39145118179183427584, 1881294510800399083201, 95950279080398196834304, 5176039012712211526485147
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A000111, A385306.
Cf. A380017, A385305, A385309, A385311.
Cf. A007559, A007788, A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
a(n) ~ n! / (sqrt(2) * Gamma(1/3) * n^(2/3) * arcsin(1/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 28 2025

A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

Original entry on oeis.org

1, 3, 9, 30, 117, 516, 2493, 13152, 75177, 460272, 3003921, 20806176, 152114013, 1169842368, 9435180357, 79553524224, 699531782481, 6400932102912, 60820145019801, 599036357936640, 6105903392066373, 64309189153428480, 698936466350352717, 7828833281592926208, 90270159223293364473
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2022

Keywords

Links

Crossrefs

Cf. A027710, A080527, A107403.
Cf. A003724, A136630, A352279.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Exp[3 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

Formula

E.g.f.: exp( 3 * sinh(x) ).
a(n) = Sum_{k=0..n} 3^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A352638 Expansion of e.g.f. 1/(1 - 3*sin(x)).

Original entry on oeis.org

1, 3, 18, 159, 1872, 27543, 486288, 10016619, 235798272, 6244714443, 183756215808, 5947907121879, 210026879004672, 8034293365747743, 330982609573398528, 14609181655918083939, 687820834029346947072, 34407546247054875367443, 1822450167175258689896448
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Cf. A000111, A007289.
Cf. A107403, A352640.
Cf. A136630.

Programs

  • Mathematica
    With[{m = 17}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*Sin[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*sin(x))))
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / (2^(3/2) * arcsin(1/3)^(n+1)). - Vaclav Kotesovec, Mar 26 2022
a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

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.

A381210 Expansion of e.g.f. 1/(1 - sinh(x))^3.

Original entry on oeis.org

1, 3, 12, 63, 408, 3123, 27552, 275103, 3065088, 37682883, 506606592, 7392091743, 116329479168, 1963781841843, 35395627487232, 678401549017983, 13776623985819648, 295481239628640003, 6674320861079273472, 158364407589097613823, 3937958237874411798528
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Crossrefs

Cf. A377530, A381207, A381209, A381211.
Cf. A136630.

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, (k+2)!*a136630(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * A136630(n,k).

A381211 Expansion of e.g.f. 1/(1 - sin(x))^3.

Original entry on oeis.org

1, 3, 12, 57, 312, 1923, 13152, 98697, 805632, 7102563, 67233792, 679970937, 7315786752, 83421156003, 1004860895232, 12749105088777, 169926064668672, 2373678328434243, 34676591077097472, 528758667342524217, 8400613520498491392, 138830752520282729283
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Crossrefs

Cf. A377530, A381207, A381209, A381210.
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, (k+2)!*I^(n-k)*a136630(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * i^(n-k) * A136630(n,k), where i is the imaginary unit.
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