A355348 G.f.: Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^(3*n-3), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

2, -7, 0, 26, -42, 63, -111, 90, 54, -273, 451, -396, 275, -561, 1287, -1781, 1365, -351, -871, 2938, -5733, 7008, -5172, 2331, -1905, 5835, -14688, 24752, -27455, 19278, -7684, -561, 10251, -32317, 69768, -104652, 107407, -72960, 31293, -10621, 18069, -63783

Offset: 0

Author

Paul D. Hanna, Jul 28 2022

Keywords

• sign

Examples

			G.f.: A(x) = 2 - 7*x + 26*x^3 - 42*x^4 + 63*x^5 - 111*x^6 + 90*x^7 + 54*x^8 - 273*x^9 + 451*x^10 - 396*x^11 + 275*x^12 - 561*x^13 + 1287*x^14 - 1781*x^15 + ...
such that
A(x) = ... + x^6/C(x)^15 + x^3/C(x)^12 + x/C(x)^9 + 1/C(x)^6 + 1/C(x)^3 + x + x^3*C(x)^3 + x^6*C(x)^6 + x^10*C(x)^9 + x^15*C(x)^12 + ... + x^(n*(n+1)/2) * C(x)^(3*n-3) + ...
also
A(x) = 1/C(x)^6 * (1 + C(x)^3)*(1 + x/C(x)^3)*(1-x) * (1 + x*C(x)^3)*(1 + x^2/C(x)^3)*(1-x^2) * (1 + x^2*C(x)^3)*(1 + x^3/C(x)^3)*(1-x^3) * (1 + x^3*C(x)^3)*(1 + x^4/C(x)^3)*(1-x^4) * ... * (1 + x^(n-1)*C(x)^3)*(1 + x^n/C(x)^3)*(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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..2555

Crossrefs

Cf. A355341, A355345, A000108.

Programs

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^(3*m-3) ); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))

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

(2) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * (C(x)^(3*n-3) + 1/C(x)^(3*n+6)).

(3) A(x) = 1/C(x)^6 * Product_{n>=1} (1 + x^(n-1)*C(x)^3) * (1 + x^n/C(x)^3) * (1-x^n), by the Jacobi triple product identity.