A214225 -id:A214225 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, Dec 03 2011

nonn

			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, ...].

G. C. Greubel, Table of n, a(n) for n = 0..370
Nantel Bergeron, Laura Colmenarejo, Shu Xiao Li, John Machacek, Robin Sulzgruber, Mike Zabrocki, Adriano Garsia, Marino Romero, Don Qui, and Nolan Wallach, Super Harmonics and a representation theoretic model for the Delta conjecture, A summary of the open problem sessions of Jan 24, 2019, Representation Theory Connections to (q,t)-Combinatorics (19w5131), Banff, BC, Canada.
John Lentfer, A conjectural basis for the (1,2)-bosonic-fermionic coinvariant ring, arXiv:2406.19715 [math.CO], 2024. See p. 2.

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

Cf. A370907, A370908.

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 *)

a(n)=n!*polcoeff(1/x*serreverse(x*exp(-x+x^2*O(x^n))/cosh(x+x^2*O(x^n))),n)

a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp((n+1)*X)*cosh(X)^(n+1)/(n+1),n)

a(n)=sum(k=0,n+1,binomial(n+1,k)*k^n/(n+1)/2)

/* 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)}

/* 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

/* 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

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

A202617 E.g.f. satisfies: A(x) = exp( x*(1 + A(x)^2)/2 ).

Original entry on oeis.org

1, 1, 3, 19, 185, 2441, 40747, 823691, 19564785, 534145105, 16482667091, 567343245635, 21552042260905, 895664877901145, 40422799315249275, 1968883362773653051, 102942561775293158369, 5750760587905912310177, 341848844954020959953059, 21545207157567497255044979

Offset: 0

Paul D. Hanna, Dec 21 2011

nonn

Compare to e.g.f. F(x) of A007889: F(x) = exp(x*(1 + F(x))/2), where A007889(n) = number of intransitive (or alternating) trees: vertices are [0,n] and for no i

Related sequence: A058014(n) = number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 185*x^4/4! + 2441*x^5/5! +...
where log(A(x)) = x*(1 + A(x)^2)/2 and
log(A(x)) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + 23616*x^6/6! +...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 576*x^4/4! + 7872*x^5/5! + 134656*x^6/6! +...

Seiichi Manyama, Table of n, a(n) for n = 0..372

Cf. A007889, A058014, A214225, A138860, A202357.

CoefficientList[Series[Sqrt[-ProductLog[-E^x*x]/x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 10 2014 *)

a(n)=local(A=1+x); for(i=0, n, A=exp(x*(1+A^2)/2 +x*O(x^n))); n!*polcoeff(A, n)

/* Coefficients of A(x)^p are given by: */
{a(n, p=1)=(1/2^n)*sum(k=0, n, binomial(n, k)*p*(2*k+p)^(n-1))}

a(n)=n!*polcoeff(exp(sum(k=1,n,k^(k-1)*cosh(k*x +x*O(x^n))*x^k/k!) +x*O(x^n)),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 20 2012

E.g.f. A(x) equals the formal inverse of function 2*log(x)/(1+x^2).
E.g.f.: exp( Sum_{n>=1} n^(n-1) * cosh(n*x) * x^n / n! ). - Paul D. Hanna, Nov 20 2012
E.g.f.: exp(G(x)) where G(x) = x/(1 - tanh(G(x))) is the e.g.f. of A214225. - Paul D. Hanna, Nov 20 2012
E.g.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the e.g.f. of A058014.
a(n) = (1/2^n)*Sum_{k=0..n} C(n,k)*(2*k+1)^(n-1).
Powers of e.g.f.:
If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(2*k+p)^(n-1).
a(n) ~ sqrt(1+c) * n^(n-1) / (2 * exp(n) * c^(n+1/2)), where c = LambertW(exp(-1)) = 0.278464542761... (see A202357). - Vaclav Kotesovec, Jan 10 2014
E.g.f.: sqrt(-LambertW(-x*exp(x))/x). - Vaclav Kotesovec, Jan 10 2014

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

Paul D. Hanna, Jul 07 2012

nonn

			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! +...

Robert Israel, Table of n, a(n) for n = 1..362

Cf. A201594, A214222, A214223, A214225.

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

m = 20; CoefficientList[InverseSeries[Series[x(1-Tan[x]), {x, 0, m}], x]/x, x] Range[m]! (* Jean-François Alcover, Apr 01 2019 *)

{a(n)=(n-1)!*polcoeff(x/(1 - tan(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))

{a(n)=n!*polcoeff(serreverse(x-x*tan(x+x*O(x^n))), n)}

{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)}

{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)}

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.

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

Paul D. Hanna, Jul 07 2012

nonn

			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! +...

Cf. A201627, A214223, A214224, A214225.

Rest[CoefficientList[InverseSeries[Series[x - x*Sin[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 12 2014 *)

{a(n)=(n-1)!*polcoeff(x/(1 - sin(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))

{a(n)=n!*polcoeff(serreverse(x-x*sin(x+x*O(x^n))), n)}

{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)}

{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)}

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

A214223 E.g.f. satisfies: A(x) = x/(1 - sinh(A(x))).

Original entry on oeis.org

1, 2, 12, 124, 1800, 33606, 766976, 20689208, 643996800, 22719618250, 895853071872, 39043448067636, 1863697724715008, 96698693656306574, 5418685033626992640, 326140667283301420912, 20983722785088536346624, 1437191703493403790787218, 104400577820040681757736960

Offset: 1

Paul D. Hanna, Jul 07 2012

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 124*x^4/4! + 1800*x^5/5! +...
Related expansions:
A(x) = x + x*sinh(x) + d/dx x^2*sinh(x)^2/2! + d^2/dx^2 x^3*sinh(x)^3/3! + d^3/dx^3 x^4*sinh(x)^4/4! +...
log(A(x)/x) = sinh(x) + d/dx x*sinh(x)^2/2! + d^2/dx^2 x^2*sinh(x)^3/3! + d^3/dx^3 x^3*sinh(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! + 109568*x^6/6! +...+ A201628(n)*x^n/n! +...
sinh(A(x)) = x + 2*x^2/2! + 13*x^3/3! + 136*x^4/4! + 1981*x^5/5! + 37056*x^6/6! + 846777*x^7/7! + 22861952*x^8/8! +...

Cf. A201628, A214222, A214224, A214225.

Rest[CoefficientList[InverseSeries[Series[x - x*Sinh[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 12 2014 *)

{a(n)=(n-1)!*polcoeff(x/(1 - sinh(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))

{a(n)=n!*polcoeff(serreverse(x-x*sinh(x+x*O(x^n))), n)}

{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*sinh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}

{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)*sinh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

E.g.f. A(x) satisfies:
(1) A(x - x*sinh(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*sinh(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*sinh(x)^n/n! ).
a(n) = n*A201628(n-1).
a(n) = (n-1)! * [x^n] x/(1 - sinh(x))^n.
a(n) ~ n^(n-1) / (sqrt(s+(2-s^2)*cosh(s)) * exp(n) * (s^2*cosh(s))^(n-1/2)), where s = 0.465767175470891411756875... is the root of the equation s*cosh(s) = 1-sinh(s). - Vaclav Kotesovec, Jan 12 2014

A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).

Original entry on oeis.org

0, 1, 8, 96, 1792, 46080, 1511424, 60325888, 2837970944, 153778913280, 9432255692800, 646039266656256, 48874810528235520, 4047655951598092288, 364221261622538141696, 35384754572803304325120, 3691411033400626898796544, 411569264258973944034361344

Offset: 0

Seiichi Manyama, Feb 11 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A100526, A216857, A360545.

Cf. A038049, A201595, A214225, A360547.

A360548 := proc(n)
    add((2*k)^(n-1)*binomial(n,k),k=1..n) ;
end proc:
seq(A360548(n),n=0..60) ; # R. J. Mathar, Mar 12 2023

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(2*x))/2)))

a(n) = sum(k=1, n, (2*k)^(n-1)*binomial(n, k));

E.g.f.: A(x) = (-1/2) * LambertW(-2*x * exp(2*x)).
a(n) = Sum_{k=1..n} (2*k)^(n-1) * binomial(n,k) = 4^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (LambertW(exp(-1))^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023

Showing 1-6 of 6 results.

cp's OEIS Frontend

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

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A202617 E.g.f. satisfies: A(x) = exp( x*(1 + A(x)^2)/2 ).

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

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

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A214223 E.g.f. satisfies: A(x) = x/(1 - sinh(A(x))).

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A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).

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