A352701 G.f. (1/x)*Series_Reversion( x*(1-x)*(3 - 2*x + sqrt(1+4*x-4*x^2))/4 ).

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

Offset

0,3

Comments

Essentially an unsigned version of A351771 (after dropping the initial term).

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

Links

Table of n, a(n) for n=0..24.

Formula

The g.f. A(x) satisfies:

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

(2) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2;

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

(4) 1 - x*(A(x) - A(-x))/2 = x/Series_Reversion( x - x*(C(x) + C(-x))/2 ), where C(x) = (1 - sqrt(1-4*x))/2 is the Catalan function (A000108);

(5a) (1/A(x) + 1/A(-x))/2 = ( 1 - x*(A(x) - A(-x))/2 )^2;

(5b) (1/A(x) - 1/A(-x))/2 = (-x)/(1 - 2*x*(A(x) - A(-x))/2);

(6a) (1/A(x)^2 + 1/A(-x)^2)/2 = ( 1 - x*(A(x) - A(-x))/2 )^3.

(6b) (1/A(x)^2 - 1/A(-x)^2)/2 = -2*x*(1 - x*(A(x) - A(-x))/2)^2/( 1 - x*(A(x) - A(-x)) ).

Let B(x) = Series_Reversion( x*(1-x^2)/(1+x^2)^3 ), then

(7) A(x) = (1 - sqrt(1 - 4*x - 4*x*B(x)^2))/(2*x);

(8) A(x) - x*A(x)^2 = A(-x) + x*A(-x)^2 = 1 + B(x)^2;

(9) 1 - x*(A(x) - A(-x))/2 = 1/(1 + B(x)^2);

(10) 1/A(x) = 1/(1 + B(x)^2)^2 - x*(1 + B(x)^2)/(1 - B(x)^2);

(10a) (1/A(x) + 1/A(-x))/2 = 1/(1 + B(x)^2)^2;

(10b) (1/A(x) - 1/A(-x))/2 = (-x)*(1 + B(x)^2)/(1 - B(x)^2);

(11) 1/A(x)^2 = 1/(1 + B(x)^2)^3 - 2*x/(1 - B(x)^4);

(11a) (1/A(x)^2 + 1/A(-x)^2)/2 = 1/(1 + B(x)^2)^3;

(11b) (1/A(x)^2 - 1/A(-x)^2)/2 = -2*x/(1 - B(x)^4).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 28*x^4 + 79*x^5 + 350*x^6 + 1075*x^7 + 5020*x^8 + 16180*x^9 + 78023*x^10 + 259417*x^11 + ...
such that A(x) = (1/x)*Series_Reversion(x*G(x)) and A(x*G(x)) = 1/G(x),
where G(x) = (1-x)*(3 - 2*x + sqrt(1+4*x-4*x^2))/4, which starts
G(x) = 1 - x - x^2 + 3*x^3 - 8*x^4 + 26*x^5 - 92*x^6 + 344*x^7 - 1336*x^8 + 5336*x^9 - 21776*x^10 + ...
Let B(x) =  Series_Reversion( x*(1-x^2)/(1+x^2)^3 ),
B(x) = x + 4*x^3 + 39*x^5 + 496*x^7 + 7180*x^9 + 112236*x^11 + 1846082*x^13 + 31485120*x^15 + ...,
then A(x) = 1 + x*A(x)^2 + B(x)^2, where
B(x)^2 = x^2 + 8*x^4 + 94*x^6 + 1304*x^8 + 19849*x^10 + 320600*x^12 + 5396108*x^14 + 93615864*x^16 + ...

Prog

(PARI) {a(n) = my(A = (1/x)*serreverse( x*(1-x)*(3 - 2*x + sqrt(1+4*x-4*x^2 +x*O(x^n) ))/4 ));
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A351771, A352383.

Sequence in context: A148754 A148755 A148756 * A351771 A148757 A148758

Adjacent sequences: A352698 A352699 A352700 * A352702 A352703 A352704

Keyword

nonn

Author

Paul D. Hanna, Mar 29 2022

Status

approved