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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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) ).

A193342 E.g.f.: A(x) = G(x)*exp(-x/2)/x where G(x) satisfies: G(G(x)) = x*exp(G(x)), and A(x) = Sum_{n>=0} a(n)*x^(2*n)/((2*n)!*4^n).

Original entry on oeis.org

1, 1, -7, 873, -335023, 314308145, -608475110391, 2176841249613401, -13293673514920102879, 130392618478782066711009, -1956708639203083689685074535, 43167469497976800185127921454793, -1354293569879914292359532215444184463, 58748391997267678043451322126451570916113
Offset: 0

Views

Author

Paul D. Hanna, Jul 23 2011

Keywords

Comments

It is surprising that the e.g.f. of this sequence is an even function.

Examples

			G.f.: A(x) = 1 + 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) - 608475110391*x^12/(12!*2^12) + 2176841249613401*x^14/(14!*2^14) +...
where G(x) = x*A(x)*exp(x/2) satisfies G(G(x)) = x*exp(G(x)):
G(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) +...
and is the e.g.f. of A193341.

Crossrefs

Cf. A193341.

Programs

  • PARI
    {a(n)=local(A=x+x^2); for(i=1, 2*n, A=A+(x*exp(A+O(x^(2*n+1)))-subst(A, x, A))/2); if(n<0,0,(2*n)!*4^n*polcoeff(A/x*exp(-x/2+O(x^(2*n+1))), 2*n))}
Showing 1-2 of 2 results.

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