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.

A276909 E.g.f. A(x) satisfies: Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)).

Original entry on oeis.org

1, 0, 3, 0, 85, 0, 6111, 0, 872649, 0, 195062395, 0, 76208072733, 0, 12330526252695, 0, 125980697776559377, 0, -857710566759117989133, 0, 11428318296234746748941925, 0, -222333914273403535165432496561, 0, 6242434914385931957857138485252825, 0, -244888574110309970555770302512462694549, 0, 13082369513456349871152908238665975845490989, 0, -930879791318792717095933863751868808486774883065, 0
Offset: 1

Views

Author

Paul D. Hanna, Sep 26 2016

Keywords

Comments

It appears that a(6*k+5) = 1 (mod 3) for k>=0 with a(n) = 0 (mod 3) elsewhere.
Apart from signs, essentially the same as A276910.
E.g.f. A(x) equals the series reversion of the e.g.f. of A276908.

Examples

			E.g.f.: A(x) = x + 3*x^3/3! + 85*x^5/5! + 6111*x^7/7! + 872649*x^9/9! + 195062395*x^11/11! + 76208072733*x^13/13! + 12330526252695*x^15/15! + 125980697776559377*x^17/17! - 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! - 222333914273403535165432496561*x^23/23! + 6242434914385931957857138485252825*x^25/25! +...
such that Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)).
RELATED SERIES.
A(x)*exp(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1446*x^6/6! + 13888*x^7/7! + 156472*x^8/8! + 2034000*x^9/9! + 29724490*x^10/10! + 476806176*x^11/11! + 8502508884*x^12/12! + 174802753216*x^13/13! + 3768345692398*x^14/14! + 63300353418240*x^15/15! + 1386349221087856*x^16/16! + 149879079531401472*x^17/17! +...+ A276911(n)*x^n/n! +...
exp(A(x)) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 116*x^5/5! + 661*x^6/6! + 8632*x^7/7! + 70617*x^8/8! + 1247248*x^9/9! + 13329001*x^10/10! + 285675776*x^11/11! + 3782734693*x^12/12! + 107823153088*x^13/13! + 1685127882621*x^14/14! + 28683829833856*x^15/15! + 574020572798641*x^16/16! + 133507199865641216*x^17/17! +...+ A276912(n)*x^n/n! +...
Also,  A( A(x)*exp(A(x)) ) = -LambertW(-x), which begins:
A( A(x)*exp(A(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! +...

Links

Crossrefs

Cf. A276910, A276911, A276912, A276913, A179270, A276908.

Programs

  • PARI
    {a(n) = my(A=x +x*O(x^n));
for(i=1,n, A = A + (x - subst(A*exp(A),x,A*exp(-A)))/2); n!*polcoeff(A,n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(V=[1], A=x); for(i=1, n\2+1, V = concat(V, [0, 0]); A = sum(m=1, #V, V[m]*x^m/m!) +x*O(x^#V); V[#V] = -(#V)!/2 * polcoeff( subst( A*exp(A), x, A*exp(-A) ), #V) ); V[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A( A(x)*exp(A(x)) ) = -LambertW(-x),
(2) A( A(x)*exp(-A(x)) ) = LambertW(x),
where LambertW( x*exp(x) ) = x.
(3) Series_Reversion( A( x*exp(x) ) ) = A( x*exp(-x) ).

A276910 E.g.f. A(x) satisfies: inverse of function A(x)*exp(i*A(x)) equals the conjugate, A(x)*exp(-i*A(x)), where i=sqrt(-1).

Original entry on oeis.org

1, 0, -3, 0, 85, 0, -6111, 0, 872649, 0, -195062395, 0, 76208072733, 0, -12330526252695, 0, 125980697776559377, 0, 857710566759117989133, 0, 11428318296234746748941925, 0, 222333914273403535165432496561, 0, 6242434914385931957857138485252825, 0, 244888574110309970555770302512462694549, 0, 13082369513456349871152908238665975845490989, 0, 930879791318792717095933863751868808486774883065, 0
Offset: 1

Views

Author

Paul D. Hanna, Sep 22 2016

Keywords

Comments

Apart from signs, essentially the same as A276909.

Examples

			E.g.f.: A(x) = x - 3*x^3/3! + 85*x^5/5! - 6111*x^7/7! + 872649*x^9/9! - 195062395*x^11/11! + 76208072733*x^13/13! - 12330526252695*x^15/15! + 125980697776559377*x^17/17! + 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! + 222333914273403535165432496561*x^23/23! + 6242434914385931957857138485252825*x^25/25! +...
such that Series_Reversion( A(x)*exp(i*A(x)) ) = A(x)*exp(-i*A(x)).
RELATED SERIES.
A(x)*exp(i*A(x)) = x + 2*I*x^2/2! - 6*x^3/3! - 28*I*x^4/4! + 180*x^5/5! + 1446*I*x^6/6! - 13888*x^7/7! - 156472*I*x^8/8! + 2034000*x^9/9! + 29724490*I*x^10/10! - 476806176*x^11/11! - 8502508884*I*x^12/12! + 174802753216*x^13/13! + 3768345692398*I*x^14/14! - 63300353418240*x^15/15! - 1386349221087856*I*x^16/16! + 149879079531401472*x^17/17! +...+ A276911(n)*i^(n-1)*x^n/n! +...
exp(i*A(x)) = 1 + I*x - x^2/2! - 4*I*x^3/3! + 13*x^4/4! + 116*I*x^5/5! - 661*x^6/6! - 8632*I*x^7/7! + 70617*x^8/8! + 1247248*I*x^9/9! - 13329001*x^10/10! - 285675776*I*x^11/11! + 3782734693*x^12/12! + 107823153088*I*x^13/13! - 1685127882621*x^14/14! - 28683829833856*I*x^15/15! + 574020572798641*x^16/16! + 133507199865641216*I*x^17/17! +...+ A276912(n)*i^(n-1)*x^n/n! +...
Also,  A( A(x)*exp(i*A(x)) ) = i*LambertW(-i*x), which begins:
A( A(x)*exp(i*A(x)) ) = x + 2*I*x^2/2! - 9*x^3/3! - 64*I*x^4/4! + 625*x^5/5! + 7776*I*x^6/6! - 117649*x^7/7! - 2097152*I*x^8/8! +...+ -n^(n-1)*(-i)^(n-1)*x^n/n! +...

Links

Crossrefs

Cf. A276909, A276911, A276912, A179270.

Programs

  • PARI
    {a(n) = my(V=[1],A=x); for(i=1,n\2+1, V = concat(V,[0,0]); A = sum(m=1,#V,V[m]*x^m/m!) +x*O(x^#V); V[#V] = -(#V)!/2 * polcoeff( subst( A*exp(I*A), x, A*exp(-I*A) ),#V) );V[n]}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: A( A(x)*exp(i*A(x)) ) = i*LambertW(-i*x), where LambertW( x*exp(x) ) = x.

A276911 E.g.f. A(x) satisfies: A(A( x*exp(-x) )) = x*exp(x).

Original entry on oeis.org

1, 2, 6, 28, 180, 1446, 13888, 156472, 2034000, 29724490, 476806176, 8502508884, 174802753216, 3768345692398, 63300353418240, 1386349221087856, 149879079531401472, 5097575010920072850, -780487993325688128000, -32524149870689487270260, 10927977097616993825596416, 490896441869732669067535414, -213936255246865273137807851520, -10450262329586550037066790750808, 6047981224337998054714885264691200
Offset: 1

Views

Author

Paul D. Hanna, Sep 22 2016

Keywords

Comments

Former name was "Inverse of e.g.f. A(x) equals its conjugate, where A(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! and i=sqrt(-1)." - Paul D. Hanna, Sep 06 2018

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1446*x^6/6! + 13888*x^7/7! + 156472*x^8/8! + 2034000*x^9/9! + 29724490*x^10/10! + ...
such that A(A( x*exp(-x) )) = x*exp(x).
RELATED SERIES.
Let F(x) = x + 2*I*x^2/2! - 6*x^3/3! - 28*I*x^4/4! + 180*x^5/5! + 1446*I*x^6/6! - 13888*x^7/7! - 156472*I*x^8/8! + 2034000*x^9/9! + 29724490*I*x^10/10! - 476806176*x^11/11! - 8502508884*I*x^12/12! + 174802753216*x^13/13! + 3768345692398*I*x^14/14! - 63300353418240*x^15/15! - 1386349221087856*I*x^16/16! + 149879079531401472*x^17/17! +...+ a(n)*i^(n-1)*x^n/n! +...
then
(a) Series_Reversion( F(x) ) = conjugate( F(x) ).
(b) F(x) = G(x)*exp(i*G(x)) where G(x) is the e.g.f. of A276910:
(c) G(x) = x - 3*x^3/3! + 85*x^5/5! - 6111*x^7/7! + 872649*x^9/9! - 195062395*x^11/11! + 76208072733*x^13/13! - 12330526252695*x^15/15! + 125980697776559377*x^17/17! + 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! +...+ A276910(n)*x^n/n! +...
where
G( F(x) ) = x + 2*I*x^2/2! - 9*x^3/3! - 64*I*x^4/4! + 625*x^5/5! + 7776*I*x^6/6! - 117649*x^7/7! - 2097152*I*x^8/8! +...+ -n^(n-1)*(-i)^(n-1)*x^n/n! +...

Links

Crossrefs

Cf. A276910, A276912.

Programs

  • PARI
    {a(n) = my(V=[1],A=x,G=x); for(i=1,n\2+1, V = concat(V,[0,0]); G = sum(m=1,#V,V[m]*x^m/m!) +x*O(x^#V);
A = G*exp(I*G); V[#V] = -(#V)!/2 * polcoeff( subst( A, x, conj(A) ),#V) ); n!*(-I)^(n-1)*polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: A(A( x*exp(-x) )) = x*exp(x). - Paul D. Hanna, Sep 06 2018
E.g.f. A(x) satisfies: A(-A(-x)) = x. - Paul D. Hanna, Sep 06 2018
Inverse of F(x) equals its conjugate, where F(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! and i=sqrt(-1).
Let G(x) be the e.g.f. of A276910, then F(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! satisfies:
(1) F(x) = G(x) * exp(i*G(x)).
(2) G( F(x) ) = i*LambertW(-i*x), where LambertW( x*exp(x) ) = x.
E.g.f. A(x) satisfies: A(A(x)) is e.g.f. of A089946 with offset 1. - Alexander Burstein, Jan 15 2022

Extensions

Name replaced with simpler formula by Paul D. Hanna, Sep 06 2018

A276913 E.g.f. A(x) satisfies: Series_Reversion( A(x)*exp(x) ) = A(x)*exp(-x).

Original entry on oeis.org

1, 0, 3, 0, -35, 0, 6111, 0, -3015207, 0, 3457389595, 0, -7910176435083, 0, 32652618744201015, 0, -225992449753641748943, 0, 2477459751096859267509171, 0, -41090881423264757483386565235, 0, 992851798453466404257942193460239, 0, -33857339246997857308988305386104611575, 0, 1586206583926227307173185697414192414735051, 0, -99763501980273385738989314186327124186627104987, 0
Offset: 1

Views

Author

Paul D. Hanna, Oct 01 2016

Keywords

Comments

It appears that a(6*k+5) = 1 (mod 3) for k>=0 with a(n) = 0 (mod 3) elsewhere.

Examples

			E.g.f.: A(x) = x + 3*x^3/3! - 35*x^5/5! + 6111*x^7/7! - 3015207*x^9/9! + 3457389595*x^11/11! - 7910176435083*x^13/13! + 32652618744201015*x^15/15! - 225992449753641748943*x^17/17! + 2477459751096859267509171*x^19/19! - 41090881423264757483386565235*x^21/21! + 992851798453466404257942193460239*x^23/23! - 33857339246997857308988305386104611575*x^25/25! +...
RELATED SERIES.
By definition, Series_Reversion( A(x)*exp(x) ) = A(x)*exp(-x), where
A(x)*exp(x) = x + 2*x^2/2! + 6*x^3/3! + 16*x^4/4! - 144*x^6/6! + 5488*x^7/7! + 47104*x^8/8! - 2799360*x^9/9! - 29427200*x^10/10! + 3293554176*x^11/11! + 40830142464*x^12/12! - 7642645477376*x^13/13! - 109489995819008*x^14/14! + 31826754503424000*x^15/15! +...+ A193341(n)*x^n/n! +...
A(x)*exp(-x) = x - 2*x^2/2! + 6*x^3/3! - 16*x^4/4! + 144*x^6/6! + 5488*x^7/7! - 47104*x^8/8! - 2799360*x^9/9! +...+ (-1)^(n-1)*A193341(n)*x^n/n! +...
Also, A( A(x)*exp(x) ) = x*exp( A(x)*exp(x) ), where
A( A(x)*exp(x) ) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 325*x^5/5! + 2046*x^6/6! + 14749*x^7/7! + 166664*x^8/8! + 1855305*x^9/9! - 8673830*x^10/10! - 380002799*x^11/11! + 33613835388*x^12/12! + 913029698893*x^13/13! - 91462474379626*x^14/14! - 2893000394547675*x^15/15! + 452208618208709776*x^16/16! +...
exp( A(x)*exp(x) ) = 1 + x + 3*x^2/2! + 13*x^3/3! + 65*x^4/4! + 341*x^5/5! + 2107*x^6/6! + 20833*x^7/7! + 206145*x^8/8! - 867383*x^9/9! - 34545709*x^10/10! + 2801152949*x^11/11! + 70233053761*x^12/12! - 6533033884259*x^13/13! - 192866692969845*x^14/14! + 28263038638044361*x^15/15! +...
Also,
A'( A(x)*exp(-x) ) * exp( A(x)*exp(-x) ) = exp(x)/(A'(x) - A(x)) - x, or
x*A'( A(x)*exp(-x) ) / A( A(x)*exp(-x) ) = exp(x)/(A'(x) - A(x)) - x.
The series reversion begins:
Series_Reversion( A(x) ) = x - 3*x^3/3! + 125*x^5/5! - 19551*x^7/7! + 8072217*x^9/9! - 7563307675*x^11/11! + 14604702539349*x^13/13! - 53272560312696375*x^15/15! + 338351296939319691953*x^17/17! +...

Links

Crossrefs

Cf. A276909, A276908, A276912, A193341.

Programs

  • PARI
    {a(n) = my(A = x +x*O(x^n));for(i=1,n, A = A + (x - subst(A*exp(x +x*O(x^n)),x, A*exp(-x +x*O(x^n))))/2); n!*polcoeff(A,n)}
for(n=1, 31, print1(a(n), ", "))

Formula

E.g.f. A(x) also satisfies:
(1) A( A(x)*exp(x) ) = x*exp( A(x)*exp(x) ).
(2) A( A(x)*exp(-x) ) = x*exp( -A(x)*exp(-x) ).
Showing 1-4 of 4 results.

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