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.

A276912 E.g.f.: A(x) satisfies: Series_Reversion( log(A(x)) * A(x) ) = log(A(x)) / A(x).

Original entry on oeis.org

1, 1, 1, 4, 13, 116, 661, 8632, 70617, 1247248, 13329001, 285675776, 3782734693, 107823153088, 1685127882621, 28683829833856, 574020572798641, 133507199865641216, 2477747434090344913, -832289494713919714304, -16453576647394853560899, 11260772482520581109810176, 246622016098219255086463333, -219530418791080092679815129088, -5247252347909156791432867741559, 6177525915951437030555334153342976, 160073955175697692672876432040185401
Offset: 0

Views

Author

Paul D. Hanna, Sep 24 2016

Keywords

Examples

			E.g.f.: 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! +...
such that Series_Reversion( log(A(x)) * A(x) ) = log(A(x)) / A(x).
RELATED SERIES.
The logarithm of the e.g.f. is an odd function:
log(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! +...+ i^(n-1)*A276910(n)*x^n/n! +...
and thus A(x) = 1/A(-x).
log(A(x)) * 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! +...+ A276911(n)*x^n/n! +...
log(A(x)) / 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! +...+ (-1)^(n-1)*A276911(n)*x^n/n! +...
RELATION TO LambertW(x):
A( log(A(x)) * A(x) ) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! +...+ (n+1)^(n-1)*x^n/n! +...
which equals LambertW(-x) / (-x).
A( log(A(x)) / A(x) ) = 1 + x - x^2/2! + 4*x^3/3! - 27*x^4/4! + 256*x^5/5! - 3125*x^6/6! + 46656*x^7/7! +...+ (n-1)^(n-1)*(-x)^n/n! +...
which equals x / LambertW(x).

Links

Crossrefs

Cf. A276910, A276911.

Programs

  • PARI
    {a(n) = my(A=1+x,L); for(i=1,n, L = log(A +x*O(x^n)); A = exp( sqrt( L*A* serreverse(L*A) ) ) ); n!*polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

E.g.f. A(x) also satisfies:
(1) A( log(A(x)) / A(x) ) = x / LambertW(x).
(2) A( log(A(x)) * A(x) ) = LambertW(-x) / (-x).
(3) A(x) * A(-x) = 1.

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) ).

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
Showing 1-3 of 3 results.

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