A215880 -id:A215880 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, Aug 25 2012

nonn

			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)! +...

G. C. Greubel, Table of n, a(n) for n = 0..385

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

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

CoefficientList[Series[Sqrt[-LambertW[-x]/LambertW[x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

{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)}

{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)}

{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),", "))

x='x+O('x^30); Vec(serlaplace(sqrt(-lambertw(-x)/lambertw(x)))) \\ G. C. Greubel, Feb 19 2018

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

A138734 a(n) = n-th term of the (n+1)-th inverse binomial transform of A138737 for n>=0.

Original entry on oeis.org

1, 0, 4, 0, 176, 0, 24192, 0, 6966528, 0, 3459768320, 0, 2642804895744, 0, 2880385956446208, 0, 4248075487709560832, 0, 8150896954121845211136, 0, 19738484719911211593891840, 0, 58891248341123034677673721856, 0

Offset: 0

Paul D. Hanna, Apr 05 2008

nonn

The n-th term of the n-th inverse binomial transform of A138737 = (n+1)^(n-1) for n>=0.

			E.g.f.: A(x) = 1 + 4*x^2/2! + 176*x^4/4! + 24192*x^6/6! + 6966528*x^8/8! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A138737, A215880.

CoefficientList[Series[-LambertW[x]*LambertW[-x]/x^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=(k+1)^(k-1)-polcoeff(subst(Ser(A),x,x/(1+k*x+x*O(x^k)))/(1+k*x),k)); polcoeff(subst(Ser(A),x,x/(1+(n+1)*x+x*O(x^n)))/(1+(n+1)*x),n)}

{a(n)=sum(k=0, n, (-1)^k*binomial(n,k)*(k+1)^(k-1)*(n-k+1)^(n-k-1))}

{a(n)=local(LW=sum(m=1,n+1,m^(m-1)*x^m/m!)+x^2*O(x^n));n!*polcoeff(-LW*subst(LW,x,-x)/x^2,n)}

{a(n)=n!*polcoeff(exp(sum(m=1,n,2*(2*m)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n)),n)}

x='x+O('x^50); Vec(serlaplace(-lambertw(x)*lambertw(-x)/x^2)) \\ G. C. Greubel, Nov 15 2017

From Paul D. Hanna, Apr 20 2011: (Start)
a(n) = Sum_{k=0..n} (-1)^k*C(n,k) * (k+1)^(k-1) * (n-k+1)^(n-k-1).
E.g.f.: -LambertW(x)*LambertW(-x)/x^2.
E.g.f.: exp( Sum_{n>=1} 2*(2n)^(2n-1) * x^(2n)/(2n)! ). (End)
Let 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)!
then e.g.f. A(x) = Chw(x)^2 - Shw(x)^2. - Paul D. Hanna, Aug 22 2012
If n is even, a(n) ~ 2*exp(2) * LambertW(exp(-1)) * n^(n-1). - Vaclav Kotesovec, Oct 08 2013

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

Paul D. Hanna, Aug 25 2012

nonn

			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)! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A215880, A215881, A138737, A215890.

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

CoefficientList[Series[-LambertW[-x]/LambertW[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

{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)}

{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)}

{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),", "))

x='x+O('x^30); Vec(serlaplace(-lambertw(-x)/lambertw(x))) \\ G. C. Greubel, Feb 19 2018

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

A215890 E.g.f.: Shw(x) / Chw(x), where Chw(x) = Sum_{n>=0} (2n+1)^(2n-1) * x^(2n)/(2n)! and Shw(x) = Sum_{n>=0} (2n+2)^(2n) * x^(2n+1)/(2n+1)!, with zero coefficients omitted.

Original entry on oeis.org

1, 7, 461, 84827, 30648697, 18319873199, 16364889804485, 20422587185959363, 33936856584735107441, 72442485189323515166807, 193177550094285124372564285, 629325007854898466696826469931, 2459394825443160188158170494692841, 11355850594904678701281528241280254207, 61160283986030409241528644478489460162357, 380011322281573634548774673447708463134715539

Offset: 1

Paul D. Hanna, Aug 25 2012

nonn

It appears that terms are not divisible by 2 or 3.

			E.g.f.: A(x) = x + 7*x^3/3! + 461*x^5/5! + 84827*x^7/7! + 30648697*x^9/9! +...
such that A(x) = Shw(x)/Chw(x) = tanh(x*Chw(x)) where
Shw(x) = x + 16*x^3/3! + 1296*x^5/5! + 262144*x^7/7! + 100000000*x^9/9! +...+ (2*n+2)^(2*n)*x^(2*n+1)/(2*n+1)! +...
Chw(x) = 1 + 3*x^2/2! + 125*x^4/4! + 16807*x^6/6! + 4782969*x^8/8! + 2357947691*x^10/10! +...+ (2*n+1)^(2*n-1)*x^(2*n)/(2*n)! +...
and Chw(x) + Shw(x) = LambertW(-x)/(-x).

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A215880, A215881, A215882, A216143.

max = 28; Chw[x_] := (-ProductLog[-x] + ProductLog[x])/(2*x); Shw[x_] := -(ProductLog[-x] + ProductLog[x])/(2*x); se = Series[ Shw[x]/Chw[x], {x, 0, max}]; (CoefficientList[se, x] // DeleteCases[#, 0] &)*Range[1, max, 2]! (* Jean-François Alcover, Jun 24 2013 *)

{a(n)=local(Chw=sum(m=0, n, (2*m+1)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^(2*n)),Shw=sum(m=0, n, (2*m+2)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^(2*n+1))); (2*n+1)!*polcoeff(Shw/Chw, 2*n+1)}
for(n=0,20,print1(a(n),", "))

{a(n)=(2*n+1)!*polcoeff( tanh( sum(m=0, n, (2*m+1)^(2*m)*x^(2*m+1)/(2*m+1)!) +x*O(x^(2*n+1))), 2*n+1)}
for(n=0,20,print1(a(n),", "))

E.g.f.: tanh(x*Chw(x)) = tanh( Sum_{n>=0} (2*n+1)^(2*n)*x^(2*n+1)/(2*n+1)! ).

A216143 E.g.f.: exp( Shw(x) ) where Shw(x) = Sum_{n>=0} (2n+2)^(2n) * x^(2n+1)/(2n+1)!.

Original entry on oeis.org

1, 1, 1, 17, 65, 1457, 10657, 307841, 3403521, 121414689, 1810995009, 77157569073, 1453708980033, 72128854709329, 1644987113677793, 93235988902015009, 2498605538747794433, 159345372352540823361, 4909485778021467744897, 348042700926255242296657

Offset: 0

Paul D. Hanna, Sep 02 2012

nonn

			E.g.f.: A(x) = 1 + x + x^2/2! + 17*x^3/3! + 65*x^4/4! + 1457*x^5/5! +...
such that A(x) = exp( Shw(x) ) where
Shw(x) = x + 16*x^3/3! + 1296*x^5/5! + 262144*x^7/7! + 100000000*x^9/9! +...+ (2*n+2)^(2*n)*x^(2*n+1)/(2*n+1)! +...
Related expansions:
Chw(x) = 1 + 3*x^2/2! + 125*x^4/4! + 16807*x^6/6! + 4782969*x^8/8! +...+ (2*n+1)^(2*n-1)*x^(2*n)/(2*n)! +...
where Chw(x) + Shw(x) = LambertW(-x)/(-x).
Also, A(x)^x is an even function:
A(x)^x = 1 + 2*x^2/2! + 76*x^4/4! + 9816*x^6/6! + 2731408*x^8/8! +...+ A215880(2*n)*x^(2*n)/(2*n)! +....

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A215880, A215881.

max = 19; Shw[x_] := -(ProductLog[-x] + ProductLog[x])/(2*x); se = Series[Exp[Shw[x]] , {x, 0, max}]; (CoefficientList[se, x] // DeleteCases[#, 0] &)*Range[0, max]! (* Jean-François Alcover, Jun 24 2013 *)

{a(n)=n!*polcoeff(exp(sum(m=0, n,(2*m+2)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)), n)}

{a(n)=local(LW=sum(m=1, n+2,-(-1)^m*m^(m-1)*x^m/m!)+x^4*O(x^n)); n!*polcoeff( (-subst(LW, x, -x)*LW/x^2)^(1/(2*x)), n)}
for(n=0,21,print1(a(n),", "))

E.g.f.: ( -LambertW(-x)*LambertW(x)/x^2 )^(1/(2*x)).

a(n) ~ c * n^(n-1), where c = -1/2*exp(1-exp(1))*(LambertW(exp(-1))^(-exp(1)/2)-LambertW(exp(-1))^(exp(1)/2)*exp(2*exp(1))) = 3.11403283... if n is even, and c = 1/2*exp(1-exp(1))*(LambertW(exp(-1))^(-exp(1)/2)+LambertW(exp(-1))^(exp(1)/2)*exp(2*exp(1))) = 4.13355253... if n is odd. - Vaclav Kotesovec, Jul 08 2013

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

Paul D. Hanna, Sep 06 2012

nonn

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

			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) + ...

Paul D. Hanna, Table of n, a(n) for n = 1..201

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

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 *)

{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

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

Name changed and entry revised by Paul D. Hanna, Mar 24 2025

cp's OEIS Frontend

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

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A138734 a(n) = n-th term of the (n+1)-th inverse binomial transform of A138737 for n>=0.

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

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A215890 E.g.f.: Shw(x) / Chw(x), 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)!, with zero coefficients omitted.

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A216143 E.g.f.: exp( Shw(x) ) where Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.

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

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A215890 E.g.f.: Shw(x) / Chw(x), where Chw(x) = Sum_{n>=0} (2n+1)^(2n-1) * x^(2n)/(2n)! and Shw(x) = Sum_{n>=0} (2n+2)^(2n) * x^(2n+1)/(2n+1)!, with zero coefficients omitted.

A216143 E.g.f.: exp( Shw(x) ) where Shw(x) = Sum_{n>=0} (2n+2)^(2n) * x^(2n+1)/(2n+1)!.