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A249787 E.g.f. A(x) satisfies: (A(x)^5 - 10*x)^2 = (2 - A(x)^2)^5. 6
1, 1, -1, 0, -3, 96, -621, 0, -17703, 1161216, -13961241, 0, -1125897003, 111588212736, -1962807307461, 0, -313129783869903, 41521527606214656, -961674168773599281, 0, -255030903352871048403, 42355944224989145726976, -1217055203596100484316701, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

E.g.f.: (1 + 2*Series_Reversion(G(x)))^(1/2), where G(x) = ((1+2*x)^(5/2) - (1-2*x)^(5/2))/10  = x + Sum_{n>=1} x^(2*n+1)/(2*n+1)! * Product_{k=0..n-1} (4*n-1)*(4*n-3).

E.g.f.: sqrt(1 - S(x)^2) + S(x) where S(x) = Series_Reversion(x - 4*x^5/5) so that S(x) = Sum_{n>=0} (4/5)^n * (5*n)!/n! * x^(4*n+1)/(4*n+1)!.

E.g.f. A(x) satisfies:

(1) A(x)^2 + A(-x)^2 = 2.

(2) A(x)^5 - A(-x)^5 = 10*x.

(3) (A(x) - A(-x))/2 = Series_Reversion(x - 4*x^5/5).

(4) B0^2 + B1^2 = 1 where B0 = (A(x) + A(-x))/2 and B1 = (A(x) - A(-x))/2.

(5) x = (A(x)^5 - (2 - A(x)^2)^(5/2))/10.

a(4*n+3) = 0 for n>=0.

EXAMPLE

E.g.f.: A(x) = 1 + x - x^2/2! - 3*x^4/4! + 96*x^5/5! - 621*x^6/6! - 17703*x^8/8! + 1161216*x^9/9! - 13961241*x^10/10! +...

where

A(x)^2 = 1 + 2*x - 6*x^3/3! + 162*x^5/5! - 12726*x^7/7! + 1931202*x^9/9! - 489966246*x^11/11! + 184547004642*x^13/13! - 97585822900566*x^15/15! +...

A(x)^5 = 1 + 5*x + 15*x^2/2! - 195*x^4/4! + 9315*x^6/6! - 1049895*x^8/8! + 207604215*x^10/10! - 64920527595*x^12/12! + 29066153231115*x^14/14! +...

Thus the coefficients of even powers of x in A(x)^2 equal zero:

[1, 2, 0, -6, 0, 162, 0, -12726, 0, 1931202, 0, -489966246, 0, ...],

while the coefficients of odd powers of x in A(x)^5 equal zero:

[1, 5, 15, 0, -195, 0, 9315, 0, -1049895, 0, 207604215, 0, ...],

after a few initial terms.

EXPLICIT FORMULA.

Let G(x) = ((1+2*x)^(5/2) - (1-2*x)^(5/2))/10, which begins

G(x) = x + 3*x^3/3! + 3*3*x^5/5! + 3*3*35*x^7/7! + 3*3*35*99*x^9/9! + 3*3*35*99*195*x^11/11! +...+ [Product_{k=0..n-1} (4*k-1)*(4*k-3)] * x^(2*n+1)/(2*n+1)! +...

then (A(x)^2 - 1)/2 = Series_Reversion(G(x)).

ALTERNATE SERIES REVERSION FORMULA.

Let S(x) = Series_Reversion(x - 4*x^5/5):

S(x) = x + 96*x^5/5! + 1161216*x^9/9! + 111588212736*x^13/13! + 41521527606214656*x^17/17! + 42355944224989145726976*x^21/21! +...+ (4/5)^n * (5*n)!/n! * x^(4*n+1)/(4*n+1)! +...

then e.g.f. A(x) = S(x) + sqrt(1 - S(x)^2), where

sqrt(1 - S(x)^2) = 1 - x^2/2! - 3*x^4/4! - 621*x^6/6! - 17703*x^8/8! +...

PROG

(PARI) /* Explicit formula: */

{a(n)=local(A, X=x+x^2*O(x^n), S=serreverse(X - 4*x^5/5));

A = S + sqrt(1 - S^2); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* Explicit formula: */

{a(n)=local(A, X=x+x^2*O(x^n), G=((1+2*X)^(5/2) - (1-2*X)^(5/2))/10);

A=(1 + 2*serreverse(G))^(1/2); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* Formula using series expansion: */

{a(n)=local(A, G=x + sum(m=1, n\2+1, x^(2*m+1)/(2*m+1)!*prod(k=0, m-1, (4*k-1)*(4*k-3)) +x^2*O(x^n)));

A=(1 + 2*serreverse(G))^(1/2); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* Alternating zero coefficients in A(x)^2 and A(x)^5: */

{a(n)=local(A=[1, 1], E=1, M); for(i=1, n, A=concat(A, 0); M=#A;

E=sum(m=0, M-1, A[m+1]*x^m/m!)+x*O(x^M);

A[M]=if(M%2==0, -(M-1)!*Vec(E^5/5)[M], -(M-1)!*Vec(E^2/2)[M])); A[n+1]}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A249788 (dual), A249785, A249786, A249923, A249922.

Sequence in context: A093009 A273442 A322460 * A264548 A094085 A300424

Adjacent sequences:  A249784 A249785 A249786 * A249788 A249789 A249790

KEYWORD

sign

AUTHOR

Paul D. Hanna, Nov 14 2014

STATUS

approved

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Last modified September 27 03:21 EDT 2020. Contains 337380 sequences. (Running on oeis4.)