A393173 - OEIS

%I #15 Mar 15 2026 10:48:20

%S 1,1,9,368,33591,5229144,1220168698,395808157536,169851873309147,

%T 93059236181644616,63367698927059812470,52500116486855196173376,

%U 52012329173814493130713494,60732567617413437556336610352,82559597601436362480749598337140,129284189427431298746234070629812288

%N G.f. A(x) satisfies [x^n] 1/(A(x) - n^2*x) = 0 for n >= 1.

%C Conjecture: for n > 0, a(n) is odd iff n is a power of 2.

%C Conjecture: a(n) == 2 (mod 4) iff n/2 is a sum of two distinct powers of 2 (A018900).

%H Paul D. Hanna, <a href="/A393173/b393173.txt">Table of n, a(n) for n = 0..512</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) [x^n] 1/(A(x) - n^2*x) = 0 for n >= 1.

%F (2) [x^n] 1/A(x) = -Sum_{k=1..n} n^(2*k) * [x^(n-k)] 1/A(x)^(k+1) for n >= 1.

%F a(n) ~ (1 - exp(-2))^2 * n^(2*n). - _Vaclav Kotesovec_, Mar 15 2026

%e G.f.: A(x) = 1 + x + 9*x^2 + 368*x^3 + 33591*x^4 + 5229144*x^5 + 1220168698*x^6 + 395808157536*x^7 + 169851873309147*x^8 + ...

%e where [x^n] 1/(A(x) - n^2*x) = 0 for n >= 1.

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in 1/(A(x) - n^2*x) begins

%e   n = 0: [1, -1,   -8,  -351,  -32800, -5156650, ...];

%e   n = 1: [1,  0,   -9,  -368,  -33510, -5222520, ...];

%e   n = 2: [1,  3,    0,  -395,  -35880, -5434002, ...];

%e   n = 3: [1,  8,   55,     0,  -37030, -5814352, ...];

%e   n = 4: [1, 15,  216,  2737,       0, -5837130, ...];

%e   n = 5: [1, 24,  567, 13024,  265050,        0, ...];

%e   n = 6: [1, 35, 1216, 41877, 1408280, 42060590, 0, ...]; ...

%e in which the main diagonal, the coefficient of x^n in row n, is all zeros after the initial term.

%o (PARI) {a(n) = my(A=[1], m); for(i=1,n, A = concat(A,0); m=#A-1;

%o A[#A] = polcoef(1/(Ser(A) - m^2*x), m) ); A[n+1]}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A393170.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 05 2026