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 1-3 of 3 results.

A201595 E.g.f. satisfies A(x) = exp(x*A(x)) * cosh(x*A(x)).

Original entry on oeis.org

1, 1, 4, 28, 288, 3936, 67328, 1385728, 33372160, 921118720, 28677169152, 994360565760, 38007586684928, 1587878686621696, 71990467473965056, 3520403893852831744, 184707311409882464256, 10350444842488122310656, 616975843658373414256640, 38981881007475178476666880
Offset: 0

Views

Author

Paul D. Hanna, Dec 03 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! +...
The coefficients of x^n/n! in initial powers of G(x) = (1 + exp(2*x))/2 begin:
G^1: [(1), 1, 2, 4, 8, 16, 32, 64, 128, ...];
G^2: [1,(2), 6, 20, 72, 272, 1056, 4160, ...];
G^3: [1, 3,(12), 54, 264, 1368, 7392, 41184, ...];
G^4: [1, 4, 20,(112), 680, 4384, 29600, 207232, ...];
G^5: [1, 5, 30, 200,(1440), 11000, 88080, 732800, ...];
G^6: [1, 6, 42, 324, 2688,(23616), 217392, 2080224, ...];
G^7: [1, 7, 56, 490, 4592, 45472,(471296), 5076400, ...];
G^8: [1, 8, 72, 704, 7344, 80768, 928512,(11085824), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 112/4, 1440/5, 23616/6, 471296/7, 11085824/8, ...].

Links

Crossrefs

Cf. A214225, A201594, A074932, A218300, A218301, A218302.
Cf. A370907, A370908.

Programs

  • Mathematica
    Join[{1},Table[Sum[Binomial[n+1,k] k^n/(n+1),{k,0,n+1}]/2,{n,20}]] (* Harvey P. Dale, Feb 04 2012 *)
CoefficientList[Series[(x-LambertW[-x*E^x])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 04 2012 *)
  • PARI
    a(n)=n!*polcoeff(1/x*serreverse(x*exp(-x+x^2*O(x^n))/cosh(x+x^2*O(x^n))),n)
  • PARI
    a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp((n+1)*X)*cosh(X)^(n+1)/(n+1),n)
  • PARI
    a(n)=sum(k=0,n+1,binomial(n+1,k)*k^n/(n+1)/2)
  • PARI
    /* Formula for a(n,m) where A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!: */
{a(n,m=1)=sum(k=0, n+m, binomial(n+m, k)*k^n*m/(n+m)/2^m)}
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(A=sum(k=0,n,(k+1)^(k-1)*cosh((k+1)*x+x*O(x^n))*x^k/k!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Oct 24 2012
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(A=1+sum(k=1,n,k^k*sinh(k*x+x^2*O(x^n))/(k*x)*x^k/k!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Nov 20 2012

Formula

a(n) = (1/2) * Sum_{k=0..n+1} C(n+1,k) * k^n / (n+1).
a(n) = [x^n/n!] exp((n+1)*x) * cosh(x)^(n+1) / (n+1).
E.g.f. A(x) satisfies:
(1) A( x*exp(-x)/cosh(x) ) = exp(x)*cosh(x).
(2) A(x) = (1/x)*Series_Reversion( x*exp(-x)/cosh(x) ).
(3) A(x) = (1 + exp(2*x*A(x)))/2.
(4) A(x) = exp(G(x)) where G(x) is the e.g.f. of A074932.
(5) A(x) = Sum_{n>=0} (n+1)^(n-1) * cosh((n+1)*x) * x^n/n!. - Paul D. Hanna, Oct 24 2012
(6) A(x) = 1 + Sum_{n>=1} n^n * sinh(n*x)/(n*x) * x^n/n!. - Paul D. Hanna, Nov 20 2012
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = Sum_{k=0..n+m} C(n+m, k) * k^n * m/(n+m) / 2^m.
a(n) = A214225(n+1)/(n+1).
E.g.f.: (x-LambertW(-x*exp(x)))/(2*x). - Vaclav Kotesovec, Dec 04 2012
a(n) ~ n!*sqrt(LambertW(exp(-1))+1)/(2*sqrt(2*Pi)*n^(3/2)*LambertW(exp(-1))^(n+1)). - Vaclav Kotesovec, Dec 04 2012
G.f.: 1/2 + 1/2 * Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1). - Seiichi Manyama, Apr 23 2024
a(n) = n! * Sum_{k=0..n} 2^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024

A201627 E.g.f. satisfies: A(x) = 1/(1 - sin(x*A(x))).

Original entry on oeis.org

1, 1, 4, 29, 312, 4481, 80768, 1754549, 44647040, 1303097665, 42923116032, 1575332861101, 63754405679104, 2820829737123841, 135469202252333056, 7018336152909163205, 390175030207597805568, 23169468447962190613121, 1463683656780476860989440, 98016257612539018485477821
Offset: 0

Views

Author

Paul D. Hanna, Dec 03 2011

Keywords

Comments

Coefficients in the expansion of 1/(1-sin(x)) yield the Euler numbers (A000111).

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 29*x^3/3! + 312*x^4/4! + 4481*x^5/5! +...
The coefficients in initial powers of G(x) = 1/(1 - sin(x)) begin:
G^1: [(1), 1, 2, 5, 16, 61, 272, 1385, 7936, ..., A000111(n+1), ...];
G^2: [1,(2), 6, 22, 96, 482, 2736, 17302, ...];
G^3: [1, 3,(12), 57, 312, 1923, 13152, 98697, ...];
G^4: [1, 4, 20,(116), 760, 5524, 44000, 380516, ...];
G^5: [1, 5, 30, 205,(1560), 13025, 118080, 1153105, ...];
G^6: [1, 6, 42, 330, 2856,(26886), 272832, 2963850, ...];
G^7: [1, 7, 56, 497, 4816, 50407, (565376), 6754097, ...];
G^8: [1, 8, 72, 712, 7632, 87848, 1078272,(14036392), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 116/4, 1560/5, 26886/6, 565376/7, 14036392/8, ...].

Crossrefs

Cf. A214222, A201594, A000111.

Programs

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

Formula

E.g.f. A(x) satisfies: A( x*(1 - sin(x)) ) = 1/(1 - sin(x)).
E.g.f.: (1/x)*Series_Reversion( x*(1 - sin(x)) ).
a(n) = [x^n] 1/(1 - sin(x))^(n+1) / (n+1).
a(n) = A214222(n+1)/(n+1).

A214224 E.g.f. satisfies: A(x) = x/(1 - tan(A(x))).

Original entry on oeis.org

1, 2, 12, 128, 1920, 37056, 874496, 24395776, 785387520, 28658483200, 1168842842112, 52692107132928, 2601710674640896, 139635670319366144, 8094064830515773440, 503939620849307353088, 33539757103898142179328, 2376284247629812872511488, 178564437032337539449487360
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! + 128*x^4/4! + 1920*x^5/5! +...
Related expansions:
A(x) = x + x*tan(x) + d/dx x^2*tan(x)^2/2! + d^2/dx^2 x^3*tan(x)^3/3! + d^3/dx^3 x^4*tan(x)^4/4! +...
log(A(x)/x) = tan(x) + d/dx x*tan(x)^2/2! + d^2/dx^2 x^2*tan(x)^3/3! + d^3/dx^3 x^3*tan(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 32*x^3/3! + 384*x^4/4! + 6176*x^5/5! + 124928*x^6/6! +...+ A201594(n)*x^n/n! +...
tan(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 152*x^4/4! + 2296*x^5/5! + 44496*x^6/6! + 1052848*x^7/7! + 29425024*x^8/8! +...

Links

Crossrefs

Cf. A201594, A214222, A214223, A214225.

Programs

  • Maple
    f:= b*(1-tan(b))-x:
newt:= unapply(b-normal(f/diff(f,b)),b):
B:= x:
for n from 1 to 5 do
  B:= convert(series(newt(B),x,2^n+1),polynom)
od:
seq(coeff(B,x,j)*j!, j=1..2^5); # Robert Israel, Feb 04 2019
  • Mathematica
    m = 20; CoefficientList[InverseSeries[Series[x(1-Tan[x]), {x, 0, m}], x]/x, x] Range[m]! (* Jean-François Alcover, Apr 01 2019 *)
  • PARI
    {a(n)=(n-1)!*polcoeff(x/(1 - tan(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-x*tan(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*tan(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)*tan(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*tan(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*tan(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*tan(x)^n/n! ).
a(n) = n*A201594(n-1).
a(n) = (n-1)! * [x^n] x/(1 - tan(x))^n.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.