A351771 Given g.f. A(x), the even bisections of both A(x) and A(x)^2 are equal, and the odd bisections of both A(x)^2 and A(x)^3 are equal (after the initial terms).

1, 1, -1, 3, -7, 28, -79, 350, -1075, 5020, -16180, 78023, -259417, 1278340, -4343642, 21740636, -75065787, 380161308, -1328887420, 6792111260, -23975385148, 123448657904, -439228736887, 2275311657814, -8148868193557, 42427160829508, -152792221834364

Offset

0,4

Comments

a(2*n+1) = A352383(n) for n >= 0.

Links

Table of n, a(n) for n=0..26.

Formula

G.f. A(x) satisfies:

(1a) [x^(2*n)] A(x) = [x^(2*n)] A(x)^2 for n >= 1.

(1b) [x^(2*n+1)] A(x)^2 = [x^(2*n+1)] A(x)^3 for n >= 1.

(1c) [x^(2*n+1)] A(x)^3 = [x^(2*n+1)] A(x)^4 for n >= 1.

(2) (A(x) - A(-x))/2 = x/(A(x)*A(-x)).

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

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

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

(5a) A(x)^2 = 2*x + (A(x)^2 + A(-x)^2)/2 + (A(x) - A(-x))^3/2.

(5b) A(x)^3 = 3*x + (A(x)^3 + A(-x)^3)/2 + (A(x) - A(-x))^3/2.

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

(6) A(x)^4 - A(x)^3 = x + x*(A(x) - A(-x)).

(7) A(-x) = (A(x)^2 + sqrt(A(x)^4 - 8*x*A(x)))/(2*A(x)).

(8) (A(x) - A(-x))^3/2 = 4*x*F(x^2), where F(x) = Series_Reversion( x*(1+x)^3/(1+2*x)^6 ).

(9) A(x)^2 - A(x) = Series_Reversion( x - x*(C(x) + C(-x))/2 ), where C(x) = x + C(x)^2 is the Catalan power series (A000108).

(10) A(x) = 1 + Series_Reversion( x*(1+x)*(3 + 2*x + sqrt(1-4*x-4*x^2))/4 ).

(11) 0 = 2*x^2 + A(x)*(1 - A(x))*(1 + 2*A(x))*x + A(x)^4*(1 - A(x))^2.

Example

G.f. A(x) = 1 + x - 1*x^2 + 3*x^3 - 7*x^4 + 28*x^5 - 79*x^6 + 350*x^7 - 1075*x^8 + 5020*x^9 - 16180*x^10 + 78023*x^11 + ...
Compare A(x) with the coefficients in the following series expansions:
A(x)^2 = 1 + 2*x - 1*x^2 + 4*x^3 - 7*x^4 + 36*x^5 - 79*x^6 + 444*x^7 - 1075*x^8 + 6324*x^9 - 16180*x^10 + 97872*x^11 + ...
A(x)^3 = 1 + 3*x + 0*x^2 + 4*x^3 - 3*x^4 + 36*x^5 - 40*x^6 + 444*x^7 - 579*x^8 + 6324*x^9 - 9000*x^10 + 97872*x^11 + ...
A(x)^4 = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 36*x^5 + 16*x^6 + 444*x^7 + 121*x^8 + 6324*x^9 + 1040*x^10 + 97872*x^11 + ...
which illustrate the properties that the coefficients of x^k for even k in A(x) and A(x)^2 are equal, and that the coefficients of x^k for odd k > 1 in A(x)^2, A(x)^3, and A(x)^4 are equal.
Related series.
(1) Notice that A(x)^2 - A(x) forms an odd function:
A(x)^2 - A(x) = x + x^3 + 8*x^5 + 94*x^7 + 1304*x^9 + 19849*x^11 + 320600*x^13 + 5396108*x^15 + ...
such that the series reversion begins
Series_Reversion( A(x)^2 - A(x) ) = x - x^3 - 5*x^5 - 42*x^7 - 429*x^9 - 4862*x^11 - 58786*x^13 - ...
which equals x - x*(C(x) + C(-x))/2, where C(x) = x + C(x)^2:
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ...
and is the Catalan power series C(x) = (1 - sqrt(1-4*x))/2.
(2) Also, the coefficients in the following series form a bisection of A(x):
(A(x)^4 - A(x)^3 - x)/2 = x^2 + 3*x^4 + 28*x^6 + 350*x^8 + 5020*x^10 + 78023*x^12 + 1278340*x^14 + ... + A352383(n)*x^(2*n+2) + ...
(3) Further, a series bisection of A(x)^2, A(x)^3, and A(x)^4 is
(A(x) - A(-x))^3/2 = 4*x^3 + 36*x^5 + 444*x^7 + 6324*x^9 + 97872*x^11 + 1598940*x^13 + 27136744*x^15 + ... + 4*A352384(n)*x^(2*n+3) + ...
which is equal to 4*x*F(x^2), where F( x*(1+x)^3/(1+2*x)^6 ) = x, and
F(x) = x + 9*x^2 + 111*x^3 + 1581*x^4 + 24468*x^5 + 399735*x^6 + 6784186*x^7 + ... + A352384(n)*x^(n+1) + ...
with
(F(x)/x)^(1/3) = 1 + 3*x + 28*x^2 + 350*x^3 + 5020*x^4 + 78023*x^5 + 1278340*x^6 + ... + A352383(n)*x^n + ...
(4) The above observations lead to the composition of functions
Series_Reversion(A(x) - 1) = [x - x*(C(x) + C(-x))/2] o (x + x^2)
which is equivalent to
Series_Reversion(A(x) - 1) = x*(1+x)*(3 + 2*x + sqrt(1-4*x-4*x^2))/4.

Mathematica

CoefficientList[1 + InverseSeries[Series[x*(1 + x)*(3 + 2*x + Sqrt[1 - 4*x - 4*x^2])/4, {x, 0, 30}], x], x] (* Vaclav Kotesovec, Mar 15 2022 *)

Prog

(PARI) /* Using Series Reversion */
{a(n) = my(A = 1 + serreverse( x*(1+x)*(3 + 2*x + sqrt(1-4*x-4*x^2 +x^2*O(x^n)))/4)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From [x^(2*n)] A(x) - A(x)^2 = 0 and [x^(2*n+1)] A(x)^2 - A(x)^3 = 0  */
{a(n) = my(A = 1 + x +x^2*O(x^n));
for(k=2, n, if(k%2==0,
A = A + x^k*polcoeff(A^1 - A^2, k),
A = A + x^k*polcoeff(A^2 - A^3, k)));
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A352383, A352384, A000108.

Sequence in context: A148755 A148756 A352701 * A148757 A148758 A054277

Adjacent sequences: A351768 A351769 A351770 * A351772 A351773 A351774

Keyword

sign

Author

Paul D. Hanna, Mar 14 2022

Status

approved