A249786 E.g.f. A(x) satisfies: (A(x)^2 - 4*x)^3 = (2 - A(x)^3)^2.

1, 1, -2, 6, -48, 360, -4800, 58800, -1088640, 18627840, -440294400, 9699782400, -278672486400, 7519473561600, -254211897139200, 8123999659776000, -315817889587200000, 11668326078689280000, -512656874530504704000, 21503534793369108480000, -1053509824992697712640000

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

E.g.f.: (1 + 3*Series_Reversion(G(x)))^(1/3), where G(x) = ((1+3*x)^(2/3) - (1-3*x)^(2/3))/4 = x + Sum_{n>=1} x^(2*n+1)/(2*n+1)! * Product_{k=0..n-1} (6*k+1)*(6*k+4).

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

(1) A(x)^3 + A(-x)^3 = 2.

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

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

a(n) ~ (-1)^(n+1) * 2^(4*n/3-1/6) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 15 2014

Example

E.g.f.: A(x) = 1 + x - 2*x^2/2! + 6*x^3/3! - 48*x^4/4! + 360*x^5/5! - 4800*x^6/6! + 58800*x^7/7! - 1088640*x^8/8! + 18627840*x^9/9! - 440294400*x^10/10! +...
where
A(x)^2 = 1 + 2*x - 2*x^2/2! - 24*x^4/4! - 1680*x^6/6! - 295680*x^8/8! - 97977600*x^10/10! - 52583731200*x^12/12! - 41661536716800*x^14/14! +...
A(x)^3 = 1 + 3*x - 12*x^3/3! - 360*x^5/5! - 40320*x^7/7! - 9797760*x^9/9! - 4151347200*x^11/11! - 2717056742400*x^13/13! - 2542118971392000*x^15/15! +...
Thus the coefficients of odd powers of x in A(x)^2 equal zero:
[1, 2, -2, 0, -24, 0, -1680, 0, -295680, 0, -97977600, 0, -52583731200, 0,...],
while the coefficients of even powers of x in A(x)^3 equal zero:
[1, 3, 0, -12, 0, -360, 0, -40320, 0, -9797760, 0, -4151347200, 0, ...],
after a few initial terms.
EXPLICIT FORMULA.
Let G(x) = ((1+3*x)^(2/3) - (1-3*x)^(2/3))/4, which begins
G(x) = x + 4*x^3/3! + 4*70*x^5/5! + 4*70*208*x^7/7! + 4*70*208*418*x^9/9! + 4*70*208*418*700*x^11/11! +...+ [Product_{k=0..n-1} (6*k+1)*(6*k+4)]*x^(2*n+1)/(2*n+1)! +...
then (A(x)^3 - 1)/3 = Series_Reversion(G(x)).
The coefficients in G(x) form triple factorials (A007559) that begin:
[1, 0, 4, 0, 280, 0, 58240, 0, 24344320, 0, 17041024000, 0, ...].

Prog

(PARI) /* Explicit formula: */
{a(n)=local(A, X=x+x^2*O(x^n), G=((1+3*X)^(2/3) - (1-3*X)^(2/3))/4);
A=(1 + 3*serreverse(G))^(1/3); 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, (6*k+1)*(6*k+4)) +x^2*O(x^n)));
A=(1 + 3*serreverse(G))^(1/3); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Alternating zero coefficients in A(x)^2 and A(x)^3: */
{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^3/3)[M])); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A249785 (dual), A249787, A249788, A249789.

Sequence in context: A052586 A052554 A228159 * A292934 A195203 A365285

Adjacent sequences: A249783 A249784 A249785 * A249787 A249788 A249789

Keyword

sign

Author

Paul D. Hanna, Nov 13 2014

Status

approved