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 1-4 of 4 results.

A215880 Expansion of e.g.f.: sqrt( -LambertW(x)*LambertW(-x)/x^2 ).

Original entry on oeis.org

1, 0, 2, 0, 76, 0, 9816, 0, 2731408, 0, 1327394080, 0, 998915378880, 0, 1076892609950080, 0, 1575062807865569536, 0, 3002359129582191616512, 0, 7232182908954405203184640, 0, 21483641086491531479896545280, 0, 77155816749532379835853517131776, 0
Offset: 0

Views

Author

Paul D. Hanna, Aug 25 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x^2/2! + 76*x^4/4! + 9816*x^6/6! + 2731408*x^8/8! + ...
such that A(x) = sqrt( -LambertW(x)*LambertW(-x)/x^2 ) where
LambertW(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 625*x^5/5! - 7776*x^6/6! + 117649*x^7/7! - 2097152*x^8/8! + ... + (-n)^(n-1)*x^n/n! + ...
Related expansions:
A(x)^2 = 1 + 4*x^2/2! + 176*x^4/4! + 24192*x^6/6! + 6966528*x^8/8! + 3459768320*x^10/10! + ... + A138734(n)*x^n/n! + ...
log(A(x)) = 2*x^2/2! + 64*x^4/4! + 7776*x^6/6! + 2097152*x^8/8! + 1000000000*x^10/10! + ... + (2*n)^(2*n-1)*x^(2*n)/(2*n)! + ...

Links

Crossrefs

Cf. A215881, A215882, A138734, A216143, A215890, A216409.

Programs

  • GAP
    List([0..25],n->Sum([0..n],k->(-1)^k*Binomial(n,k)*(k+(1/2))^(k-1)*(n-k+(1/2))^(n-k-1)/4)); # Muniru A Asiru, Feb 19 2018
  • Mathematica
    CoefficientList[Series[Sqrt[-LambertW[x]*LambertW[-x]/x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
  • PARI
    {a(n)=local(LW=sum(m=1, n+1,-(-1)^m*m^(m-1)*x^m/m!)+x^2*O(x^n)); n!*polcoeff(sqrt(-LW*subst(LW, x, -x)/x^2), n)}
  • PARI
    {a(n)=local(Chw=sum(m=0, n\2, (2*m+1)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n),Shw=sum(m=0, n\2, (2*m+2)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)); n!*polcoeff(sqrt(Chw^2-Shw^2), n)}
  • PARI
    /* E.g.f.: sqrt(Chw(x,t)^2 - Shw(x,t)^2)^(1/t) for any |t|>0: */
{a(n)=local(Chw_t=sum(m=0, n\2, t*(2*m+t)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n),
Shw_t=sum(m=0, n\2, t*(2*m+t+1)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n));
n!*polcoeff(sqrt(Chw_t^2-Shw_t^2)^(1/t), n)}
  • PARI
    {a(n)=n!*polcoeff(exp(sum(m=1, n, (2*m)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n)), n)}
  • PARI
    {a(n)=sum(k=0, n, (-1)^k*binomial(n, k)*(k+1/2)^(k-1)*(n-k+1/2)^(n-k-1)/4)}
for(n=0,21,print1(a(n),", "))
  • PARI
    x='x+O('x^30); Vec(serlaplace(sqrt(-lambertw(x)*lambertw(-x)/ x^2))) \\ G. C. Greubel, Feb 19 2018

Formula

E.g.f.: exp( Sum_{n>=1} (2*n)^(2*n-1) * x^(2*n)/(2*n)! ).
E.g.f.: sqrt( Chw(x)^2 - Shw(x)^2 ), where
Chw(x) = Sum_{n>=0} (2*n+1)^(2*n-1) * x^(2*n)/(2*n)! and
Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.
E.g.f.: sqrt( Chw(x,t)^2 - Shw(x,t)^2 )^(1/t) for |t|>0, where
Chw(x,t) = Sum_{n>=0} t*(2*n+t)^(2*n-1) * x^(2*n)/(2*n)! and
Shw(x,t) = Sum_{n>=0} t*(2*n+t+1)^(2*n) * x^(2*n+1)/(2*n+1)!.
a(n) = Sum_{k=0..n} (-1)^k*C(n,k) * (k + 1/2)^(k-1) * (n-k + 1/2)^(n-k-1) / 4.
If n is even, a(n) ~ exp(1)*sqrt(LambertW(exp(-1)))*n^(n-1) = 1.434430245088497558... * n^(n-1). - Vaclav Kotesovec, Nov 27 2012

A215881 Expansion of e.g.f.: sqrt( -LambertW(-x) / LambertW(x) ).

Original entry on oeis.org

1, 1, 1, 10, 37, 716, 4741, 136760, 1314377, 50468752, 637409641, 30580648352, 479025538861, 27578021183168, 515932095998957, 34657964676194176, 754078761294069649, 57902855910383448320, 1436649321508321044817, 124128617507138965088768, 3459197142121422461242421
Offset: 0

Views

Author

Paul D. Hanna, Aug 25 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 10*x^3/3! + 37*x^4/4! + 716*x^5/5! + 4741*x^6/6! +...
such that A(x) = sqrt( -LambertW(-x)/LambertW(x) ) where
LambertW(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 625*x^5/5! - 7776*x^6/6! + 117649*x^7/7! - 2097152*x^8/8! +...+ (-n)^(n-1)*x^n/n! +...
Related expansions:
log(A(x)) = x + 9*x^3/3! + 625*x^5/5! + 117649*x^7/7! + 43046721*x^9/9! +...+ (2*n-1)^(2*n-2)*x^(2*n-1)/(2*n-1)! +...

Links

Crossrefs

Cf. A215880, A215882, A215890, A138734, A216143.

Programs

  • GAP
    List([0..25],n->Sum([0..n],k->-(-1)^k*Binomial(n,k)*(k-(1/2))^(k-1)*(n-k+(1/2))^(n-k-1)/4)); # Muniru A Asiru, Feb 19 2018
  • Mathematica
    CoefficientList[Series[Sqrt[-LambertW[-x]/LambertW[x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
  • PARI
    {a(n)=local(LW=sum(m=1, n+1,-(-1)^m*m^(m-1)*x^m/m!)+x^2*O(x^n)); n!*polcoeff(sqrt(-subst(LW, x, -x)/LW), n)}
  • PARI
    {a(n)=n!*polcoeff(exp(sum(m=0, n, (2*m+1)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)), n)}
  • PARI
    {a(n)=sum(k=0, n, -(-1)^k*binomial(n, k)*(k-1/2)^(k-1)*(n-k+1/2)^(n-k-1)/4)}
for(n=0,21,print1(a(n),", "))
  • PARI
    x='x+O('x^30); Vec(serlaplace(sqrt(-lambertw(-x)/lambertw(x)))) \\ G. C. Greubel, Feb 19 2018

Formula

E.g.f.: exp( Sum_{n>=0} (2*n+1)^(2*n) * x^(2*n+1)/(2*n+1)! ).
a(n) = Sum_{k=0..n} -(-1)^k*C(n,k) * (k - 1/2)^(k-1) * (n-k + 1/2)^(n-k-1) / 4.
a(n) ~ c * n^(n-1), where c = 1/2*(1-LambertW(exp(-1))) / sqrt(LambertW(exp(-1))) = 0.6836640292259232... if n is even and c = 1/2*(1+LambertW(exp(-1))) / sqrt(LambertW(exp(-1))) = 1.2113614261884947... if n is odd. - Vaclav Kotesovec, Nov 27 2012

A215882 Expansion of e.g.f.: -LambertW(-x) / LambertW(x).

Original entry on oeis.org

1, 2, 4, 26, 160, 2002, 21184, 395866, 5980160, 149083874, 2933576704, 91549564570, 2222207205376, 83345185392562, 2407376957456384, 105482963294851418, 3534260251308064768, 177194291803516980418, 6757029862401745616896, 381514700506253250858778
Offset: 0

Views

Author

Paul D. Hanna, Aug 25 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 26*x^3/3! + 160*x^4/4! + 2002*x^5/5! +... such that A(x) = -LambertW(-x)/LambertW(x) where LambertW(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 625*x^5/5! - 7776*x^6/6! + 117649*x^7/7! - 2097152*x^8/8! +...+ (-n)^(n-1)*x^n/n! +... .
Related expansions: log(A(x)) = 2*x + 18*x^3/3! + 1250*x^5/5! + 235298*x^7/7! + 86093442*x^9/9! +...+ 2*(2*n+1)^(2*n)*x^(2*n+1)/(2*n+1)! +...

Links

Crossrefs

Cf. A215880, A215881, A138737, A215890.

Programs

  • GAP
    List([0..20],n->Sum([0..n],k->-(-1)^k*Binomial(n,k)*(k-1)^(k-1)*(n-k+1)^(n-k-1))); # Muniru A Asiru, Feb 20 2018
  • Mathematica
    CoefficientList[Series[-LambertW[-x]/LambertW[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
  • PARI
    {a(n)=local(LW=sum(m=1, n+1,-(-1)^m*m^(m-1)*x^m/m!)+x^2*O(x^n)); n!*polcoeff(sqrt(-subst(LW, x, -x)/LW), n)}
  • PARI
    {a(n)=n!*polcoeff(exp(sum(m=0, n, 2*(2*m+1)^(2*m)*x^(2*m+1)/ (2*m+1)!)+x*O(x^n)), n)}
  • PARI
    {a(n)=sum(k=0, n, -(-1)^k*binomial(n, k)*(k-1)^(k-1)*(n-k+1)^(n-k-1))}
for(n=0,21,print1(a(n),", "))
  • PARI
    x='x+O('x^30); Vec(serlaplace(-lambertw(-x)/lambertw(x))) \\ G. C. Greubel, Feb 19 2018

Formula

E.g.f.: exp( Sum_{n>=0} 2*(2*n+1)^(2*n) * x^(2*n+1)/(2*n+1)! ).
a(n) = Sum_{k=0..n} -(-1)^k*C(n,k) * (k-1)^(k-1) * (n-k+1)^(n-k-1).
a(n) ~ c * n^(n-1), where c = (1-LambertW(exp(-1))^2)/LambertW(exp(-1)) = 3.31265693390754834... if n is even and c = (1+LambertW(exp(-1))^2)/ LambertW(exp(-1)) = 3.86958601942969593... if n is odd. - Vaclav Kotesovec, Nov 27 2012

A216409 Expansion of e.g.f. (x/cos(x)) * exp(x*tan(x)) (odd powers only).

Original entry on oeis.org

1, 9, 185, 6769, 384849, 31247161, 3421948361, 485057489505, 86270172949025, 18789108183911401, 4913945007420622425, 1518613513007413125073, 547156929866111948071025, 227227144424871839232479769, 107701858026047543489146771049
Offset: 1

Views

Author

Paul D. Hanna, Sep 06 2012

Keywords

Comments

Name changed Mar 24 2025; prior name was: e.g.f. Series_Reversion( x*Cw(x) ), which is equivalent to formula (2).

Examples

			E.g.f.: A(x) = x + 9*x^3/3! + 185*x^5/5! + 6769*x^7/7! + 384849*x^9/9! + ...
Related expansions.
Define Cw(x) and Sw(x) by Cw(x) + i*Sw(x) = LambertW(-i*x)/(-i*x), then
Cw(x) + i*Sw(x) = (cos(x*Cw(x)) + i*sin(x*Cw(x))) * exp(-x*Sw(x)) and
Cw(x) = 1 - 3*x^2/2! + 125*x^4/4! - 16807*x^6/6! + 4782969*x^8/8! -+ ... + (-1)^n*(2*n+1)^(2*n-1)*x^(2*n)/(2*n)! + ...
where A(x*Cw(x)) = x, and
Sw(x) = x - 16*x^3/3! + 1296*x^5/5! - 262144*x^7/7! + 100000000*x^9/9! -+ ... + (-1)^n*(2*n+2)^(2*n)*x^(2*n+1)/(2*n+1)! + ...
where A( arctan(Sw(x)/Cw(x)) ) = x.
Also, Sw(x)/Cw(x) = tan( x*Cw(x) ) and begins
Sw(x)/Cw(x) = x - 7*x^3/3! + 461*x^5/5! - 84827*x^7/7! + 30648697*x^9/9! - 18319873199*x^11/11! +- ... + (-1)^(n-1) * A215890(n)*x^(2*n-1) + ...

Links

Crossrefs

Cf. A215890, A138734, A215880, A215881, A215882, A216143.

Programs

  • Mathematica
    nmax = 20; Quiet[Rest[CoefficientList[Series[x*E^(Sqrt[x]*Tan[Sqrt[x]])*Sec[Sqrt[x]], {x, 0, nmax}], x] * (2*Range[0, nmax]-1)!]] (* Vaclav Kotesovec, Mar 24 2025 *)
  • PARI
    {a(n)=local(Cw=sum(m=0,n,(-1)^m*(2*m+1)^(2*m-1)*x^(2*m)/(2*m)!) +x*O(x^n));n!*polcoeff(serreverse(x*Cw),n)}
for(n=1,20,print1(a(2*n-1),", ")) \\ print only odd-indexed terms

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1)/(2*n-1)! satisfies:
(1) Sum_{n>=0} (-1)^n * (2*n+1)^(2*n) * A(x)^(2*n+1)/(2*n+1)! = x.
(2) A( x*Cw(x) ) = x where Cw(x) = Sum_{n>=0} (-1)^n * (2*n+1)^(2*n-1) * x^(2*n)/(2*n)!.
(3) A( arctan(Sw(x)/Cw(x)) ) = x where Sw(x) = Sum_{n>=0} (-1)^n * (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.
(4) A(x) = (x/cos(x)) * exp(x*tan(x)). - Paul D. Hanna, Mar 24 2025
a(n) ~ 2^(4*n - 5/4) * n^(2*n - 3/4) / (Pi^(2*n-1) * exp(2*n - 2*sqrt(2*n) + 1/2)) * (1 - (14 + Pi^2)/(12*sqrt(2*n))). - Vaclav Kotesovec, Mar 24 2025

Extensions

Name changed and entry revised by Paul D. Hanna, Mar 24 2025
Showing 1-4 of 4 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.