OFFSET
0,2
COMMENTS
Conjecture 1: a(k) is odd iff k = 2^n - 2 for n >= 1.
Conjecture 2: a(2^n - 2) == 3 (mod 16) for n > 1.
Is there a closed formula for the g.f. of this sequence? Compare to the g.f. of A365516.
Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (A000108):
(1) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1.
(2) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1.
Related identity: F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1), which holds formally for all Maclaurin series F(x). - Paul D. Hanna, Oct 03 2023
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..600
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+2)^(n-2) for n > 1.
(2) [x^(n-1)] (1 + (n-2)*x*A(x))^n / A(x)^n = -2*n*(n-4)^(n-2) for n > 1.
(3) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = 2*n*((n+1)^(n-2) - (n-2)^(n-2))/3 for n > 1.
(4) A(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n+2)*x*A(x))^(n+1). - Paul D. Hanna, Oct 03 2023
EXAMPLE
G.f.: A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 16*x^4 + 104*x^5 + 515*x^6 + 2090*x^7 + 8170*x^8 + 34704*x^9 + 160014*x^10 + 751282*x^11 + 3479758*x^12 + ...
RELATED TABLE.
The table of coefficients of x^k in (1 + (n+1)*x*A(x))^n/A(x)^n begins:
n=1: [1, 0, 1, 0, -11, -54, -182, -594, ...];
n=2: [1, 2, 3, 2, -21, -130, -494, -1660, ...];
n=3: [1, 6, 15, 20, -18, -288, -1391, -5070, ...];
n=4: [1, 12, 58, 144, 151, -468, -3934, -17376, ...];
n=5: [1, 20, 165, 720, 1715, 1274, -8960, -60530, ...];
n=6: [1, 30, 381, 2650, 10824, 24576, 10623, -176034, ...];
n=7: [1, 42, 763, 7812, 49084, 191016, 413343, 49818, ...];
n=8: [1, 56, 1380, 19600, 176242, 1033664, 3873296, 8000000, ...]; ...
in which the main diagonal equals n*(n+2)^(n-2) for n > 1.
PROG
(PARI) /* Formula [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+2)^(n-2) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A;
A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m+2)^(m-2) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 11 2023
STATUS
approved