OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) satisfies:
(1) [x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1) for n >= 1.
(2) A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
(3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (2*x).
(4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(2*x) dx ).
a(n) ~ c * 2^n * n! * n^(3/2), where c = 0.06926688933886004638602492... - Vaclav Kotesovec, Nov 16 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 185*x^4 + 2353*x^5 + 36075*x^6 + 638115*x^7 + 12683761*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(2*n+1) begins:
n=0: [1, 1, 3, 19, 185, 2353, 36075, ...];
n=1: [1, 3, 12, 76, 705, 8595, 127680, ...];
n=2: [1, 5, 25, 165, 1490, 17506, 252050, ...];
n=3: [1, 7, 42, 294, 2632, 30016, 419454, ...];
n=4: [1, 9, 63, 471, 4239, 47295, 643017, ...];
n=5: [1, 11, 88, 704, 6435, 70785, 939312, ...];
n=6: [1, 13, 117, 1001, 9360, 102232, 1329016, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1), n >= 1,
as illustrated by
[x^1] A(x)^3 = 3 = [x^0] 3*A(x)^3 = 3*1;
[x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
[x^3] A(x)^7 = 294 = [x^2] 7*A(x)^7 = 7*42;
[x^4] A(x)^9 = 4239 = [x^3] 9*A(x)^9 = 9*471;
[x^5] A(x)^11 = 70785 = [x^4] 11*A(x)^11 = 11*6435;
[x^6] A(x)^13 = 1329016 = [x^5] 13*A(x)^13 = 13*102232; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^2) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + ... + A317352(n)*x^n + ...
where B(x)^2 = (1/x) * Series_Reversion( x/A(x)^2 ).
PROG
(PARI) /* Using A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)) */
{a(n) = my(A=1); for(i=1, n, A = 1 + x*A^2/(A - 2*x*A' + x*O(x^n)) );
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Using [x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1)*A(x)^(2*n+1) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff((x*Ser(A)^(2*(#A)-1) - Ser(A)^(2*(#A)-1)/(2*(#A)-1)), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 08 2022
STATUS
approved