OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..200
FORMULA
G.f. A(x) satisfies:
(1) A(x) = x/( G(A(x)^2) * G(-A(x)^2) ) where G(x) = 1 + x*G(x)^2.
(2) A(x) = x - x^3*A(x)^2 - 2*x^2*A(x)^3 - x^4*A(x)^5.
(3) A(x) = Series_Reversion( x*(1 + x^2)^2 / (1 - x^2) ).
(4) A(x) = x*C(x)*D(x) where
(4.a) C(x) = C(x)^2 - A(x)^2.
(4.b) D(x) = D(x)^2 + A(x)^2.
(4.c) C(x) = (1 + x*A(x))/(1 + x^2*A(x)^2).
(4.d) D(x) = (1 - x*A(x))/(1 + x^2*A(x)^2).
(4.e) C(x) = (1 + sqrt(1 + 4*A(x)^2))/2.
(4.f) D(x) = (1 + sqrt(1 - 4*A(x)^2))/2.
(4.g) C(x) = 1/G(-A(x)^2) where G(x) = 1 + x*G(x)^2.
(4.h) D(x) = 1/G(A(x)^2) where G(x) = 1 + x*G(x)^2 is the g.f. of Catalan numbers (A000108).
a(n) = Sum_{k=0..2*n-2} (-1)^k * A278880(2*n-1,k) for n>=1.
a(n) = Sum_{k=0..2*n-2} (-1)^k * (4*n-3)/((4*n-2*k-3)*(2*k+1)) * binomial(4*n-k-4, k) * binomial(2*n+k-2, 2*n-k-2). - Paul D. Hanna, Dec 08 2016
EXAMPLE
G.f.: A(x) = x - 3*x^5 + 23*x^9 - 232*x^13 + 2671*x^17 - 33247*x^21 + 435732*x^25 - 5923596*x^29 + 82761455*x^33 - 1181085841*x^37 + 17143012047*x^41 +...
such that A(x) = x*(1 - x^2*A(x)^2)/(1 + x^2*A(x)^2)^2.
RELATED SERIES.
A(x)^2 = x^2 - 6*x^6 + 55*x^10 - 602*x^14 + 7263*x^18 - 93192*x^22 + 1247636*x^26 - 17230290*x^30 + 243669007*x^34 - 3511010950*x^38 + 51361157967*x^42 +...
G.f. A(x) = x*C(x)*D(x) where
C(x) = (1 + x*A(x))/(1 + x^2*A(x)^2) = 1 + x^2 - x^4 - 4*x^6 + 7*x^8 + 33*x^10 - 68*x^12 - 344*x^14 + 767*x^16 + 4035*x^18 +...+ A243863(n)*x^(2*n) +...
D(x) = (1 - x*A(x))/(1 + x^2*A(x)^2) = 1 - x^2 - x^4 + 4*x^6 + 7*x^8 - 33*x^10 - 68*x^12 + 344*x^14 + 767*x^16 - 4035*x^18 +...+ (-1)^n*A243863(n)*x^(2*n) +...
such that C(x)^2 - A(x)^2 = C(x) and D(x)^2 + A(x)^2 = D(x).
PROG
(PARI) {a(n) = my(A=x); for(i=0, 4*n, A = x*(1 - x^2*A^2)/(1 + x^2*A^2 +x*O(x^(4*n)))^2 ); polcoeff(A, 4*n-3)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Explicit formula from triangle A278880 */
{a(n) = sum(k=0, 2*n-2, (-1)^k * (4*n-3)/((4*n-2*k-3)*(2*k+1)) * binomial(4*n-k-4, k) * binomial(2*n+k-2, 2*n-k-2) )}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Dec 01 2016
STATUS
approved