A317800 G.f. A(x) satisfies: Sum_{n>=1} ( A(x) - (-1)^n * A(-x) )^n * (-1)^n / n = 0.

1, 1, 1, 4, 10, 33, 105, 354, 1214, 4206, 14846, 52750, 189516, 686745, 2506913, 9211226, 34036230, 126426446, 471769950, 1767460752, 6645539212, 25076120890, 94937019050, 360268374124, 1369645176012, 5226326126048, 20039843858208, 76654036799842, 290534140464144, 1123489897863753, 4582416833711249, 17212665701732282, 45565498032190230

Offset

1,4

Comments

Odd terms occur at a(2^k - 1) and a(2^k - 2) for k > 1 and at a(1), while a(n) is even elsewhere (conjecture).

First negative term is a(37).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

G.f. A(x) satisfies:

(1) A(-A(-x)) = x.

(2a) Sum_{n>=1} ( A(x) - (-1)^n * A(-x) )^n * (-1)^n / n = 0.

(2b) A(A(x)) = B(x) such that Sum_{n>=1} ( x + (-1)^n * B(x) )^n / n = 0, where B(x) is the o.g.f. of A316363.

(3a) A(A(x)) = -x + 2 * Series_Reversion( x - x^2/(1 - 2*x^2) ).

(3b) A(A(x)) = x + 2 * Series_Reversion( x/sqrt(1 + 2*x^2) - x^2 )^2.

Let C = (A(x) + A(-x))/2 and S = (A(x) - A(-x))/2, then

(4a) arctanh(2*C) + log(1 - 4*S^2)/2 = 0,

(4b) 1 - 4*S^2 = (1 - 2*C)/(1 + 2*C),

(5a) S^2 = C/(1 + 2*C),

(5b) C = S^2/(1 - 2*S^2),

(6a) A(x) = S + S^2/(1 - 2*S^2),

(6b) A(x) = C + sqrt(C/(1 + 2*C)).

(7) 0 = (2*y + y^2 - y^3) - (2 - 2*y + y^2)*A(x) + (1 + y)*A(x)^2 + A(x)^3, where y = -A(-x) = Series_Reversion(A(x)).

Example

G.f. A(x) = x + x^2 + x^3 + 4*x^4 + 10*x^5 + 33*x^6 + 105*x^7 + 354*x^8 + 1214*x^9 + 4206*x^10 + ...
Let the series bisections of g.f. A(x) be denoted by
C = (A(x) + A(-x))/2 = x^2 + 4*x^4 + 33*x^6 + 354*x^8 + 4206*x^10 + ...
S = (A(x) - A(-x))/2 = x + x^3 + 10*x^5 + 105*x^7 + 1214*x^9 + 14846*x^11 + ...
then from the definition we have
0 = (2*C) - (2*S)^2/2 + (2*C)^3/3 - (2*S)^4/4 + (2*C)^5/5 - (2*S)^6/6 + (2*C)^7/7 - (2*S)^8/8 + ...
thus  arctanh(2*C) + log(1 - 4*S^2)/2 = 0,
so that  (1 - 2*C)/(1 + 2*C) = 1 - 4*S^2.
RELATED SERIES.
A(A(x)) = x + 2*x^2 + 4*x^3 + 14*x^4 + 52*x^5 + 204*x^6 + 840*x^7 + 3574*x^8 + 15588*x^9 + 69332*x^10 + ... + A316363(n)*x^n + ...
where A(A(x)) = -x + 2 * Series_Reversion( x - x^2/(1 - 2*x^2) ).

Prog

(PARI) /* From: A(x) = S + S^2/(1 - 2*S^2) and A(x) = Series_Reversion(-A(-x)) */
{a(n) = my(A=[1, 1], S); for(i=1, n, S=(x*Ser(A) - subst(x*Ser(A), x, -x))/2; A=concat(Vec( S + S^2/(1 - 2*S^2) ), 0); if(#A%2==1, A = (A + Vec( serreverse(subst(-x*Ser(A), x, -x)) ) )/2 ); ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A316363.

Sequence in context: A052373 A007716 A122948 * A357799 A149171 A149172

Adjacent sequences: A317797 A317798 A317799 * A317801 A317802 A317803

Keyword

sign

Author

Paul D. Hanna, Aug 09 2018

Status

approved