A355348 - OEIS

%I #13 Aug 03 2022 06:17:54

%S 2,-7,0,26,-42,63,-111,90,54,-273,451,-396,275,-561,1287,-1781,1365,

%T -351,-871,2938,-5733,7008,-5172,2331,-1905,5835,-14688,24752,-27455,

%U 19278,-7684,-561,10251,-32317,69768,-104652,107407,-72960,31293,-10621,18069,-63783

%N 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).

%H Paul D. Hanna, <a href="/A355348/b355348.txt">Table of n, a(n) for n = 0..2555</a>

%F 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).

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

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

%F (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.

%e 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 + ...

%e such that

%e 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) + ...

%e also

%e 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) * ...

%e where C(x) = 1 + x*C(x)^2 begins

%e 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 + ...

%o (PARI) {a(n) = my(A,C=1/x*serreverse(x-x^2 +O(x^(n+2))),M=ceil(sqrt(2*n+9)));

%o A = sum(m=-M,M, x^(m*(m+1)/2) * C^(3*m-3) ); polcoeff(A,n)}

%o for(n=0,70,print1(a(n),", "))

%Y Cf. A355341, A355345, A000108.

%K sign

%O 0,1

%A _Paul D. Hanna_, Jul 28 2022