A363559 - OEIS

%I #17 Aug 07 2023 08:04:06

%S 1,10,81,757,6561,59454,531496,4788072,43046721,387480753,3486784492,

%T 31381709148,282429556893,2541872737062,22876792457796,

%U 205891204134565,1853020188851841,16677182431460826,150094635300957591,1350851725033981380,12157665459056934471

%N Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n).

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds as a formal power series for all y.

%H Paul D. Hanna, <a href="/A363559/b363559.txt">Table of n, a(n) for n = 0..300</a>

%F The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1.a) A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n).

%F (1.b) A(x) = Sum_{n=-oo..+oo} x^n * (3 - x^n)^(2*n).

%F (2.a) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - 3*x^n)^(2*n).

%F (2.b) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + 3*x^n)^(2*n).

%F (3.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (3 + x^n)^n.

%F (3.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (-3 + x^n)^n.

%F (4.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + 3*x^n)^n.

%F (4.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - 3*x^n)^n.

%F From _Paul D. Hanna_, Aug 06 2023: (Start)

%F The following generating functions are extensions of _Peter Bala_'s formulas given in A260147.

%F (5.a) A(x^2) = Sum_{n=-oo..+oo} x^(2*n+1) * (3 + x^(2*n+1))^(2*n+1).

%F (5.b) A(x^2) = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + 3*x^(2*n+1))^(2*n+1).

%F (End)

%F a(2^n) = 9^(2^n) for n > 0 (conjecture).

%F a(p) = p*3^(p-1) + 9^p for primes p > 3 (conjecture).

%e G.f.: A(x) = 1 + 10*x + 81*x^2 + 757*x^3 + 6561*x^4 + 59454*x^5 + 531496*x^6 + 4788072*x^7 + 43046721*x^8 + 387480753*x^9 + ...

%o (PARI) {a(n) = my(A); A = sum(m=-n-1,n+1, x^m * (3 + x^m +x*O(x^n))^(2*m) ); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A260147, A363558, A363569, A363561.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 01 2023