cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

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Author

Paul D. Hanna, Sep 11 2023

Keywords

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

Examples

			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.
		

Crossrefs

Programs

  • 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), ", "))

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
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