A365574 Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+2)^(n-2) for n > 1.

1, 2, 3, 4, 16, 104, 515, 2090, 8170, 34704, 160014, 751282, 3479758, 16012684, 74362915, 350282602, 1665651094, 7952638460, 38067823370, 182874936368, 882344022104, 4274341269824, 20773195676078, 101228332620524, 494521566769160, 2421729829067636, 11886902458813596

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

Cf. A365516, A303063, A365095.

Sequence in context: A333802 A229546 A373056 * A343494 A300855 A353164

Adjacent sequences: A365571 A365572 A365573 * A365575 A365576 A365577

Keyword

nonn

Author

Paul D. Hanna, Sep 11 2023

Status

approved