A369082 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(3*A(x)^2 + A(-x)^2)/4.
1, 1, 1, 3, 4, 15, 22, 91, 140, 612, 969, 4389, 7084, 32890, 53820, 254475, 420732, 2017356, 3362260, 16301164, 27343888, 133767543, 225568798, 1111731933, 1882933364, 9338434700, 15875338990, 79155435870, 134993766600, 676196049060, 1156393243320, 5815796869995
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 4*x^4 + 15*x^5 + 22*x^6 + 91*x^7 + 140*x^8 + 612*x^9 + 969*x^10 + 4389*x^11 + 7084*x^12 + ... RELATED SERIES. We can see from the expansion of A(x)^2, which begins A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 15*x^4 + 44*x^5 + 91*x^6 + 280*x^7 + 612*x^8 + 1938*x^9 + 4389*x^10 + 14168*x^11 + 32890*x^12 + ... that the odd bisection of A(x) is derived from the even bisection of A(x)^2: (A(x) - A(-x))/2 = x + 3*x^3 + 15*x^5 + 91*x^7 + 612*x^9 + ... (A(x)^2 + A(-x)^2)/2 = 1 + 3*x^2 + 15*x^4 + 91*x^6 + 612*x^8 + ... and the even bisection of A(x) is derived from the odd bisection of A(x)^2: (A(x) + A(-x))/2 = 1 + x^2 + 4*x^4 + 22*x^6 + 140*x^8 + 969*x^10 + ... (A(x)^2 - A(-x)^2)/2 = 2*x + 8*x^3 + 44*x^5 + 280*x^7 + 1938*x^9 + ... so that (A(x) + A(-x))/2 = 1 + (1/2)*x * (A(x)^2 - A(-x)^2)/2.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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PARI
{a(n) = if(n%2==0, binomial(2*n, n/2)/(3*n/2 + 1), 3*binomial(2*n+1,n\2)/(2*n+1))} for(n=0,30, print1(a(n),", ")) -
PARI
{a(n) = my(A=1+x, B); for(i=1, n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x); A = 1 + x*(A^2 + B^2)/2 + (1/2)*x*(A^2 - B^2)/2 ); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) -
PARI
{a(n) = my(A); A = (1/x)*serreverse( (sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)) - (1 - 6*x + 4*x^2))/8 ); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = 1 + x*(3*A(x)^2 + A(-x)^2)/4.
(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + (1/2)*x*(A(x)^2 - A(-x)^2)/2.
(2.a) (A(x) + A(-x))/2 = 1 + (1/2)*x*(A(x)^2 - A(-x)^2)/2.
(2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
(2.c) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))/2).
(2.d) (A(x) + A(-x))/2 = F(x^2) where F(x) = 1 + x*F(x)^4 (cf. A002293).
(2.e) (A(x) - A(-x))/2 = x*F(x^2)^3 where F(x) = 1 + x*F(x)^4 (cf. A006632).
(3.a) A(x) = (1 - sqrt(1 - 4*x + 2*x*A(-x) + x^2*A(-x)^2))/x.
(3.b) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2))/(2*x).
(4.a) 0 = (1+4*x) - (1+6*x)*A(x) + (3+4*x)*x*A(x)^2 - 3*x^2*A(x)^3 + x^3*A(x)^4.
(4.b) x = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - (1 - 6*x*A(x) + 4*x^2*A(x)^2))/8.
(5.a) A(x) = (1/x)*Series_Reversion( (sqrt(1 + 4*x - 4*x^2) - (1 - 6*x + 4*x^2))/8 ).
(5.b) (A(x) + A(-x))/2 = (1/x)*Series_Reversion( x/C(x^2) ) = where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan sequence (A000108).
(5.c) (A(x) - A(-x))/2 = Series_Reversion( x*D(-x^2)^3 ) where D(x) = 1 + x*D(x)^3 (cf. A001764).
(6.a) a(2*n) = binomial(4*n, n)/(3*n + 1) for n >= 0.
(6.b) a(2*n+1) = 3*binomial(4*n+3,n)/(4*n+3) for n >= 0.
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