A370141 Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + A(x)) = 1 + 2*Sum_{n>=1} x^(n*(n+1)/2).

1, -2, 4, -8, 14, -22, 28, -14, -80, 420, -1430, 4128, -10798, 26176, -59114, 123442, -232240, 365888, -355616, -475892, 4318112, -17471288, 56635490, -163101656, 432173038, -1067080032, 2456709054, -5216642696, 9906435640, -15415122000, 12937725806, 33034018944, -238942986520

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..210

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

Let Q(x) = 1 + 2*Sum_{n>=1} x^(n*(n+1)/2), then

(1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + A(x)).

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

(3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + A(x))/(1 + x^n + A(x) - F(n+1)), a continued fraction.

Example

G.f.: A(x) = x - 2*x^2 + 4*x^3 - 8*x^4 + 14*x^5 - 22*x^6 + 28*x^7 - 14*x^8 - 80*x^9 + 420*x^10 - 1430*x^11 + 4128*x^12 + ...
Let Q(x) = 1 + 2*Sum_{n>=1} x^(n*(n+1)/2)
then A = A(x) satisfies
(1) Q(x) = 1 + (x + A) + (x + A)*(x^2 + A) + (x + A)*(x^2 + A)*(x^3 + A) + (x + A)*(x^2 + A)*(x^3 + A)*(x^4 + A) + (x + A)*(x^2 + A)*(x^3 + A)*(x^4 + A)*(x^5 + A) + ...
also
(2) Q(x) = 1/(1 - A) + x/((1 - A)*(1 - x*y*A)) + x^3/((1 - A)*(1 - x*y*A)*(1 - x^2*y*A)) + x^6/((1 - A)*(1 - x*y*A)*(1 - x^2*y*A)*(1 - x^3*y*A)) + x^10/((1 - A)*(1 - x*y*A)*(1 - x^2*y*A)*(1 - x^3*y*A)*(1 - x^4*y*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
(3) Q(x) = 1/(1 - (x + A)/(1 + x + A - (x^2 + A)/(1 + x^2 + A - (x^3 + A)/(1 + x^3 + A - (x^4 + A)/(1 + x^4 + A - (x^5 + A)/(1 + x^5 + A - (x^6 + A)/(1 + x^6 + A - (x^7 + A)/(1 - ...)))))))).
where
Q(x) = 1 + 2*x + 2*x^3 + 2*x^6 + 2*x^10 + 2*x^15 + 2*x^21 + ... + 2*x^(n*(n+1)/2) + ...

Prog

(PARI) {a(n, y=1) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( (sum(m=1, #A, prod(k=1, m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1, sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1)  ); H=A; A[n+1]}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A370140, A370142, A370143, A370144.

Sequence in context: A316699 A132425 A154264 * A194694 A194692 A155506

Adjacent sequences: A370138 A370139 A370140 * A370142 A370143 A370144

Keyword

sign

Author

Paul D. Hanna, Feb 14 2024

Status

approved