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.

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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) ).

A193341 E.g.f. satisfies: A(A(x)) = x*exp(A(x)), where A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)).

Original entry on oeis.org

1, 2, 6, 16, 0, -144, 5488, 47104, -2799360, -29427200, 3293554176, 40830142464, -7642645477376, -109489995819008, 31826754503424000, 518027268557111296, -221570477108873330688, -4041287223180417957888, 2438941389381370203996160, 49292069262802363796684800
Offset: 1

Views

Author

Paul D. Hanna, Jul 23 2011

Keywords

Comments

It is surprising that A(x)*exp(-x/2)/x is an even function (cf. A193342).

Examples

			E.g.f.: A(x) = x + 2*x^2/(2!*2) + 6*x^3/(3!*4) + 16*x^4/(4!*8) - 144*x^6/(6!*32) + 5488*x^7/(7!*64) + 47104*x^8/(8!*128) - 2799360*x^9/(9!*256) - 29427200*x^10/(10!*512) +...
where A(A(x)) = x*exp(A(x)) begins:
A(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 22*x^4/4! + 90*x^5/5! + 396*x^6/6! + 1918*x^7/7! + 10830*x^8/8! + 66510*x^9/9! + 325450*x^10/10! +...
The series reversion begins:
A(x)*exp(-x) = -A(-x) = x - 2*x^2/(2!*2) + 6*x^3/(3!*4) - 16*x^4/(4!*8) + 144*x^6/(6!*32) +...
so that the g.f. satisfies: -A(x)/A(-x) = exp(x).
The e.g.f. G(x) = A(x)*exp(-x/2)/x is an even function:
G(x) = 1 + x^2/(2!*2^2) - 7*x^4/(4!*2^4) + 873*x^6/(6!*2^6) - 335023*x^8/(8!*2^8) + 314308145*x^10/(10!*2^10) +...

Crossrefs

Cf. A276913, A193342.

Programs

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

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)) also satisfies:
(1) A(x) = -A(-x)*exp(x).
(2) A( A(x)/exp(x) ) = x.
(3) A(-A(-x)) = x.
(4) A(x) = x*exp(x/2)*G(x) where G(x) is the even function described by A193342.
Showing 1-2 of 2 results.

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