A354648 G.f. A(x) satisfies: -x^3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

1, 0, 0, 1, 3, 9, 22, 54, 135, 368, 1060, 3135, 9295, 27472, 81309, 242255, 728429, 2208483, 6736523, 20634196, 63410076, 195467757, 604457802, 1875053982, 5833449236, 18195767301, 56888745654, 178238369769, 559538565187, 1759796017533, 5544359742297

Offset

0,5

Links

Table of n, a(n) for n=0..30.

Formula

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

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

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

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

(4) -x^3 = 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.

Example

G.f.: A(x) = 1 + x^3 + 3*x^4 + 9*x^5 + 22*x^6 + 54*x^7 + 135*x^8 + 368*x^9 + 1060*x^10 + 3135*x^11 + 9295*x^12 + 27472*x^13 + ...
such that A = A(x) satisfies:
(1) -x^3 = ... + 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 + ...
(2) -x^3 = (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 + ...
(3) -x^3 = (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 + ...
(4) -x^3 = (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) * ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x^3 + 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]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A268650, A354647, A354649, A000716.

Sequence in context: A350105 A336511 A054442 * A192663 A114697 A032284

Adjacent sequences: A354645 A354646 A354647 * A354649 A354650 A354651

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2022

Status

approved