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 1-5 of 5 results.

A214222 E.g.f. satisfies: A(x) = x/(1 - sin(A(x))).

Original entry on oeis.org

1, 2, 12, 116, 1560, 26886, 565376, 14036392, 401823360, 13030976650, 472154276352, 18903994333212, 828807273828352, 39491616319733774, 2032038033784995840, 112293378446546611280, 6632975513529162694656, 417050432063319431036178, 27809989478829060358799360
Offset: 1

Views

Author

Paul D. Hanna, Jul 07 2012

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 116*x^4/4! + 1560*x^5/5! +...
Related expansions:
A(x) = x + x*sin(x) + d/dx x^2*sin(x)^2/2! + d^2/dx^2 x^3*sin(x)^3/3! + d^3/dx^3 x^4*sin(x)^4/4! +...
log(A(x)/x) = sin(x) + d/dx x*sin(x)^2/2! + d^2/dx^2 x^2*sin(x)^3/3! + d^3/dx^3 x^3*sin(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 29*x^3/3! + 312*x^4/4! + 4481*x^5/5! + 80768*x^6/6! +...+ A201627(n)*x^n/n! +...
sin(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 104*x^4/4! + 1381*x^5/5! + 23616*x^6/6! + 493975*x^7/7! + 12216448*x^8/8! +...

Crossrefs

Cf. A201627, A214223, A214224, A214225.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - x*Sin[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 12 2014 *)
  • PARI
    {a(n)=(n-1)!*polcoeff(x/(1 - sin(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-x*sin(x+x*O(x^n))), n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*sin(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*sin(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. A(x) satisfies:
(1) A(x - x*sin(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*sin(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*sin(x)^n/n! ).
a(n) = n*A201627(n-1).
a(n) = (n-1)! * [x^n] x/(1 - sin(x))^n.
a(n) ~ sqrt((1-t)/(2+t)) * n^(n-1) * (sqrt(1-t^2)/(1-t)^2)^n / exp(n), where t = 0.527766122670442778... is the root of the equation t = sin(sqrt((1-t)/(1+t))). - Vaclav Kotesovec, Jan 12 2014

A381181 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + sin(x)) ).

Original entry on oeis.org

1, 1, 2, 5, 8, -79, -1584, -20539, -223616, -1855295, -1736960, 435730789, 14511117312, 338965239601, 6202042886144, 71638247035109, -714560796196864, -84697775518956799, -3650903032332091392, -115829159202293866939, -2739961030150105333760, -29414406825401517785039
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Links

Crossrefs

Cf. A201627, A381180, A381182.
Cf. A162653, A381171, A381173.
Cf. A136630, A196776.

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!*binomial(n+1, k)*I^(n-k)*a136630(n, k))/(n+1);

Formula

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

A201628 E.g.f. satisfies: A(x) = 1/(1 - sinh(x*A(x))).

Original entry on oeis.org

1, 1, 4, 31, 360, 5601, 109568, 2586151, 71555200, 2271961825, 81441188352, 3253620672303, 143361363439616, 6907049546879041, 361245668908466176, 20383791705206338807, 1234336634416972726272, 79843983527411321710401, 5494767253686351671459840, 400863405346004202504321343
Offset: 0

Views

Author

Paul D. Hanna, Dec 03 2011

Keywords

Comments

The function 1/(1-sinh(x)) is the e.g.f. of A006154, where A006154(n) is the number of labeled ordered partitions of an n-set into odd parts.

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! +...
The coefficients in initial powers of G(x) = 1/(1 - sinh(x)) begin:
G^1: [(1), 1, 2, 7, 32, 181, 1232, 9787, 88832, ..., A006154(n), ...];
G^2: [1,(2), 6, 26, 144, 962, 7536, 67706, ...];
G^3: [1, 3,(12), 63, 408, 3123, 27552, 275103, ...];
G^4: [1, 4, 20,(124), 920, 7924, 77600, 850924, ...];
G^5: [1, 5, 30, 215,(1800), 17225, 185280, 2211515, ...];
G^6: [1, 6, 42, 342, 3192,(33606), 393792, 5080662, ...];
G^7: [1, 7, 56, 511, 5264, 60487, (766976), 10634911, ...];
G^8: [1, 8, 72, 728, 8208, 102248, 1395072,(20689208), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 124/4, 1800/5, 33606/6, 766976/7, 20689208/8, ...].

Crossrefs

Cf. A214223, A201627, A201595, A006154.

Programs

  • PARI
    {a(n)=n!*polcoeff(1/x*serreverse(x*(1-sinh(x+x^2*O(x^n)))),n)}
  • PARI
    {a(n)=n!*polcoeff(1/(1 - sinh(x+x*O(x^n)))^(n+1)/(n+1), n)}

Formula

E.g.f. A(x) satisfies: A( x*(1 - sinh(x)) ) = 1/(1 - sinh(x)).
E.g.f.: (1/x)*Series_Reversion( x*(1 - sinh(x)) ).
a(n) = [x^n] 1/(1 - sinh(x))^(n+1) / (n+1).
a(n) = A214223(n+1)/(n+1).

A381180 E.g.f. A(x) satisfies A(x) = 1 + sin(x*A(x)) / A(x).

Original entry on oeis.org

1, 1, 0, -1, -8, -19, 64, 1091, 7680, -1415, -650752, -8575865, -35559424, 857890021, 21380186112, 203548592651, -1615715926016, -95486152906639, -1599622990659584, -1397194164399601, 657963431581974528, 18168041375501245021, 157453907927886725120, -6059840564222790027821
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Crossrefs

Cf. A201627, A381181, A381182.
Cf. A136630, A196776.

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!*binomial(n-k+1, k)/(n-k+1)*I^(n-k)*a136630(n, k));

Formula

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

A381182 E.g.f. A(x) satisfies A(x) = 1/( 1 - A(x) * sin(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 71, 1288, 31661, 984640, 37085075, 1641305472, 83497838425, 4801347029504, 307975150996831, 21802395720298496, 1688562016007776261, 142023935786330431488, 12892154760586821775019, 1256251152910271399624704, 130793914073764385411654321, 14490427167940362294881615872
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Crossrefs

Cf. A201627, A381180, A381181.
Cf. A136630, A196776.

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!*binomial(n+2*k+1, k)/(n+2*k+1)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*k+1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
Showing 1-5 of 5 results.

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