A249788 E.g.f. A(x) satisfies: (A(x)^2 - 4*x)^5 = (2 - A(x)^5)^2.

1, 1, -4, 12, -120, 1080, -14400, 241920, -4233600, 103783680, -2408071680, 74468782080, -2230518850560, 81262621440000, -2999630643609600, 125381910682828800, -5464949027586048000, 258986473607083622400, -12924791918438400000000, 688106822684149835366400

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..360

Formula

E.g.f.: (1 + 5*Series_Reversion(G(x)))^(1/5), where G(x) = ((1+5*x)^(2/5) - (1-5*x)^(2/5))/4 = x + Sum_{n>=1} x^(2*n+1)/(2*n+1)! * Product_{k=0..n-1} (10*k+3)*(10*k+8).

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

(1) A(x)^5 + A(-x)^5 = 2.

(2) A(x)^2 - A(-x)^2 = 4*x.

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

(4) A(x) = B(x^2) + x/B(x^2), where B(x^2) = (A(x) + A(-x))/2.

a(n) ~ (-1)^(n+1) * 2^(8*n/5-3/10) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 15 2014

Example

E.g.f.: A(x) = 1 + x - 4*x^2/2! + 12*x^3/3! - 120*x^4/4! + 1080*x^5/5! - 14400*x^6/6! + 241920*x^7/7! - 4233600*x^8/8! + 103783680*x^9/9! +...
where
A(x)^2 = 1 + 2*x - 6*x^2/2! - 48*x^4/4! + 1440*x^6/6! + 1088640*x^8/8! + 500048640*x^10/10! + 254062448640*x^12/12! +...
A(x)^5 = 1 + 5*x - 120*x^3/3! + 720*x^5/5! + 302400*x^7/7! + 96163200*x^9/9! + 37362124800*x^11/11! + 17236393574400*x^13/13! +...
Thus the coefficients of odd powers of x in A(x)^2 equal zero:
[1, 2, -6, 0, -48, 0, 1440, 0, 1088640, 0, 500048640, 0, ...],
while the coefficients of even powers of x in A(x)^5 equal zero:
[1, 5, 0, -120, 0, 720, 0, 302400, 0, 96163200, 0, 37362124800, 0, ...],
after a few initial terms.
EXPLICIT FORMULA.
Let G(x) = ((1+5*x)^(2/5) - (1-5*x)^(2/5))/4, which begins
G(x) = x + 24*x^3/3! + 5616*x^5/5! + 3616704*x^7/7! + 4535346816*x^9/9! + 9360955828224*x^11/11! +...+ [Product_{k=0..n-1} (10*k+3)*(10*k+8)]*x^(2*n+1)/(2*n+1)! +...
then (A(x)^5 - 1)/5 = Series_Reversion(G(x)).

Prog

(PARI) /* Explicit formula: */
{a(n)=local(A, X=x+x^2*O(x^n), G=((1+5*X)^(2/5) - (1-5*X)^(2/5))/4);
A=(1 + 5*serreverse(G))^(1/5); 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, (10*k+3)*(10*k+8)) +x^2*O(x^n)));
A=(1 + 5*serreverse(G))^(1/5); 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^2/2)[M], -(M-1)!*Vec(E^5/5)[M])); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A249787 (dual), A249785, A249786, A249789.

Sequence in context: A155174 A053491 A285451 * A032323 A331905 A367885

Adjacent sequences: A249785 A249786 A249787 * A249789 A249790 A249791

Keyword

sign

Author

Paul D. Hanna, Nov 15 2014

Status

approved