A354646 G.f. A(x) satisfies: 1 = Sum_{n=-oo..+oo} (x + x^n)^n * (-2*A(x))^(n*(n-1)/2).

1, -1, -4, 44, 316, -22695, -769536, 156937802, 30299780744, -18827264809946, -17187430890378027, 37887447329364481223, 148620374587239353630657, -1249806569497062808351943525, -20168103472406206381500342351035, 666759209181977763318463790517458280

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

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

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

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

Example

G.f.: A(x) = 1 - x - 4*x^2 + 44*x^3 + 316*x^4 - 22695*x^5 - 769536*x^6 + 156937802*x^7 + 30299780744*x^8 - 18827264809946*x^9 - 17187430890378027*x^10 ++-- ...
such that
B(x) = Sum_{n>=1} x^n * (1 + x^(n-1))^n * (-2*A(x))^(n*(n-1)/2)
and
B(x) = -Sum_{n>=1} x^(n^2) / (1 + x^(n+1))^n * (-2*A(x))^(n*(n+1)/2),
where
B(x) = 2*x - 2*x^2 - 10*x^3 + 98*x^4 + 618*x^5 - 45552*x^6 - 1538490*x^7 + 313926892*x^8 + 60600533658*x^9 - 37654860921240*x^10 - 34374918573912040*x^11 ++-- ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(sum(m=-#A, #A, (x + x^m)^m * (-2*Ser(A))^(m*(m-1)/2) ), #A)/2); H=A; A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A260116.

Sequence in context: A074751 A129551 A202162 * A081078 A220920 A035014

Adjacent sequences: A354643 A354644 A354645 * A354647 A354648 A354649

Keyword

sign

Author

Paul D. Hanna, Jun 07 2022

Status

approved