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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.

%I A365574 #23 Oct 07 2023 22:23:46
%S A365574 1,2,3,4,16,104,515,2090,8170,34704,160014,751282,3479758,16012684,
%T A365574 74362915,350282602,1665651094,7952638460,38067823370,182874936368,
%U A365574 882344022104,4274341269824,20773195676078,101228332620524,494521566769160,2421729829067636,11886902458813596
%N 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.
%C A365574 Conjecture 1: a(k) is odd iff k = 2^n - 2 for n >= 1.
%C A365574 Conjecture 2: a(2^n - 2) == 3 (mod 16) for n > 1.
%C A365574 Is there a closed formula for the g.f. of this sequence? Compare to the g.f. of A365516.
%C A365574 Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (A000108):
%C A365574 (1) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1.
%C A365574 (2) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1.
%C A365574 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
%H A365574 Paul D. Hanna, <a href="/A365574/b365574.txt">Table of n, a(n) for n = 0..600</a>
%F A365574 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A365574 (1) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+2)^(n-2) for n > 1.
%F A365574 (2) [x^(n-1)] (1 + (n-2)*x*A(x))^n / A(x)^n = -2*n*(n-4)^(n-2) for n > 1.
%F A365574 (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.
%F A365574 (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
%e A365574 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 + ...
%e A365574 RELATED TABLE.
%e A365574 The table of coefficients of x^k in (1 + (n+1)*x*A(x))^n/A(x)^n begins:
%e A365574 n=1: [1,  0,    1,     0,    -11,     -54,    -182,    -594, ...];
%e A365574 n=2: [1,  2,    3,     2,    -21,    -130,    -494,   -1660, ...];
%e A365574 n=3: [1,  6,   15,    20,    -18,    -288,   -1391,   -5070, ...];
%e A365574 n=4: [1, 12,   58,   144,    151,    -468,   -3934,  -17376, ...];
%e A365574 n=5: [1, 20,  165,   720,   1715,    1274,   -8960,  -60530, ...];
%e A365574 n=6: [1, 30,  381,  2650,  10824,   24576,   10623, -176034, ...];
%e A365574 n=7: [1, 42,  763,  7812,  49084,  191016,  413343,   49818, ...];
%e A365574 n=8: [1, 56, 1380, 19600, 176242, 1033664, 3873296, 8000000, ...]; ...
%e A365574 in which the main diagonal equals n*(n+2)^(n-2) for n > 1.
%o A365574 (PARI) /* Formula [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+2)^(n-2) */
%o A365574 {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A;
%o A365574 A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m+2)^(m-2) ); A[n+1]}
%o A365574 for(n=0, 30, print1(a(n), ", "))
%Y A365574 Cf. A365516, A303063, A365095.
%K A365574 nonn
%O A365574 0,2
%A A365574 _Paul D. Hanna_, Sep 11 2023