login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A179270 G.f. satisfies: inverse of function A(x) + i*A(x)^2 equals the conjugate, A(x) - i*A(x)^2, where i=sqrt(-1). 5
1, 0, -1, 0, 4, 0, -21, 0, 122, 0, -758, 0, 4958, 0, -33509, 0, 233810, 0, -1641150, 0, 12364368, 0, -71807506, 0, 1354944972, 0, 33794258600, 0, 2524565441138, 0, 186642439700891, 0, 16196862324254354, 0, 1602823227559245434, 0, 179707702260054046760, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..256

FORMULA

G.f. satisfies: A( A(x) - i*A(x)^2 ) = x*Catalan(-i*x) = i*(1-sqrt(1+4*i*x))/2.

a(n)=T(n,1), T(n, m)=1/2*(m/n*binomial(2*n-m-1, n-1)*(%i)^(n+m)*(-1)^n-(sum(k=m+1..n-1, T(k, m)*sum(i=k..n, T(n, i)*binomial(k, i-k)*(-%i)^(i-k)))+sum(i=m+1..n, T(n, i)*binomial(m, i-m)*(-%i)^(i-m)))), n>m, T(n,n)=1. [Vladimir Kruchinin, Apr 30 2012]

EXAMPLE

G.f.: A(x) = x - x^3 + 4*x^5 - 21*x^7 + 122*x^9 - 758*x^11 +...

A(x)^2 = x^2 - 2*x^4 + 9*x^6 - 50*x^8 + 302*x^10 - 1928*x^12 +...

A(x) + i*A(x)^2 = x - i*x^2 - x^3 + 2*i*x^4 + 4*x^5 - 9*i*x^6 - 21*x^7 - 50*i*x^8 + 122*x^9 +...

where Series_Reversion[A(x) + i*A(x)^2] = A(x) - i*A(x)^2.

The i-th iteration of A(x) + i*A(x)^2 is a real-valued series in x, and begins:

x - x^2 + x^3 - 2*x^5 + 3*x^6 + x^7 - 38*x^8/3 + 70*x^9/3 - 2*x^10 - 266*x^11/3 + 214*x^12 - 436*x^13/3 - 469*x^14 + 12649*x^15/9 +...

PROG

(PARI) {a(n)=local(A=x+sum(k=3, n-1, a(k)*x^k)+x*O(x^n)); if(n==1, 1, if(n%2==0, 0, -polcoeff((subst(A, x, A-I*A^2)+I*subst(A, x, A-I*A^2+x*O(x^n))^2), n)/2))}

(PARI) /* Faster vectorized version: */

{oo=100; A=[1]; B=x; C=(1-sqrt(1-4*(x+x^2+x*O(x^oo))))/2; A182399=[1];

for(n=1, oo, A182399=concat(A182399, 0); B=x*Ser(A182399);

A182399[n]=Vec((B+subst(C+x*O(x^n), x, serreverse(B)))/2)[n];

A=Vec(-I*subst(x*Ser(A182399), x, I*serreverse(x+I*x^2+x^2*O(x^n))));

print1(A[n], ", "))}

(Maxima)

T(n, m):=if n=m then 1 else 1/2*(m/n*binomial(2*n-m-1, n-1)*(%i)^(n+m)*(-1)^n-(sum(T(k, m)*sum(T(n, i)*binomial(k, i-k)*(-%i)^(i-k), i, k, n), k, m+1, n-1)+sum(T(n, i)*binomial(m, i-m)*(-%i)^(i-m), i, m+1, n)));

makelist(T(n, 1), n, 1, 10); [Vladimir Kruchinin, Apr 30 2012]

CROSSREFS

Cf. A182399, A318008, A277292.

Sequence in context: A199933 A078630 A178671 * A246132 A229827 A295839

Adjacent sequences:  A179267 A179268 A179269 * A179271 A179272 A179273

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jul 06 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 21 19:08 EST 2019. Contains 319350 sequences. (Running on oeis4.)