A354654 - OEIS

%I #7 Jun 07 2022 18:58:45

%S 5,124,9300,912520,102616748,12498655200,1604505393140,

%T 213790010204692,29287693334340840,4099332312599011100,

%U 583685111605968443456,84277588096627459702860,12310921909740521584887824,1816058097888803062860159620,270156262107594683175523302780

%N G.f. A(x) satisfies: -4 = Sum_{n=-oo..oo}  (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

%H Paul D. Hanna, <a href="/A354654/b354654.txt">Table of n, a(n) for n = 0..400</a>

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

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

%F (2) -4 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).

%F (3) -4 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).

%F (4) -4 = Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n-1)*A(x)^n) * (1 - x^n*A(x)^(n-1)), by the Jacobi Triple Product identity.

%F a(n) = (-1)^(n+1) * Sum_{k=0..2*n+1} A354649(n,k)*(-4)^k, for n >= 0.

%F a(n) = Sum_{k=0..2*n+1} A354650(n,k)*4^k, for n >= 0.

%e G.f.: A(x) = 5 + 124*x + 9300*x^2 + 912520*x^3 + 102616748*x^4 + 12498655200*x^5 + 1604505393140*x^6 + 213790010204692*x^7 + 29287693334340840*x^8 + ...

%e such that A = A(x) satisfies:

%e (1) -4 = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ...

%e (2) -4 = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...

%e (3) -4 = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ...

%e (4) -4 = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ...

%o (PARI) {a(n) = my(A=[5]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff(4 + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );A[n+1]}

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

%Y Cf. A354649, A354650, A268299, A354652, A354653, A354661, A354662, A354663, A354664.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Jun 02 2022