A355864 G.f.: Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^(4*n-6), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

2, -14, 28, 26, -203, 427, -741, 1314, -1575, 264, 3201, -7953, 11308, -11440, 13364, -26403, 50479, -68549, 59956, -19930, -50743, 165880, -319635, 436575, -424830, 308193, -258570, 488410, -1122459, 2043162, -2777783, 2771340, -1946892, 726066, 746643, -3157458, 7406770

Offset

0,1

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1325

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be obtained from the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

(1) A(x) = 1/C(x)^6 * Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^(4*n).

(2) A(x) = 1/C(x)^10 * Product_{n>=1} (1 + x^(n-1)*C(x)^4) * (1 + x^n/C(x)^4) * (1-x^n), by the Jacobi triple product identity.

Example

G.f.: A(x) = 2 - 14*x + 28*x^2 + 26*x^3 - 203*x^4 + 427*x^5 - 741*x^6 + 1314*x^7 - 1575*x^8 + 264*x^9 + 3201*x^10 - 7953*x^11 + 11308*x^12 + ...
such that
A(x) = ... + x^6/C(x)^22 + x^3/C(x)^18 + x/C(x)^14 + 1/C(x)^10 + 1/C(x)^6 + x/C(x)^2 + x^3*C(x)^2 + x^6*C(x)^6 + x^10*C(x)^10 + x^15*C(x)^14 + ... + x^(n*(n+1)/2) * C(x)^(4*n-6) + ...
also
A(x) = 1/C(x)^10 * (1 + C(x)^4)*(1 + x/C(x)^4)*(1-x) * (1 + x*C(x)^4)*(1 + x^2/C(x)^4)*(1-x^2) * (1 + x^2*C(x)^4)*(1 + x^3/C(x)^4)*(1-x^3) * (1 + x^3*C(x)^4)*(1 + x^4/C(x)^4)*(1-x^4) * ... * (1 + x^(n-1)*C(x)^4)*(1 + x^n/C(x)^4)*(1-x^n) * ...
where C(x) = 1 + x*C(x)^2 begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + ... + A000108(n)*x^n + ...

Prog

(PARI) {a(n) = my(A, C=1/x*serreverse(x-x^2 +O(x^(n+2))), M=ceil(sqrt(2*n+9)));
A = sum(m=-M, M, x^(m*(m+1)/2) * C^(4*m-6) ); polcoeff(A, n)}
for(n=0, 70, print1(a(n), ", "))

Crossrefs

Cf. A355341, A355345, A355348, A000108.

Sequence in context: A330789 A031079 A364764 * A200039 A281206 A189331

Adjacent sequences: A355861 A355862 A355863 * A355865 A355866 A355867

Keyword

sign

Author

Paul D. Hanna, Aug 02 2022

Status

approved