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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).

%I A355864 #12 Aug 03 2022 06:18:43
%S A355864 2,-14,28,26,-203,427,-741,1314,-1575,264,3201,-7953,11308,-11440,
%T A355864 13364,-26403,50479,-68549,59956,-19930,-50743,165880,-319635,436575,
%U A355864 -424830,308193,-258570,488410,-1122459,2043162,-2777783,2771340,-1946892,726066,746643,-3157458,7406770
%N 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).
%H A355864 Paul D. Hanna, <a href="/A355864/b355864.txt">Table of n, a(n) for n = 0..1325</a>
%F A355864 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 A355864 (1) A(x) = 1/C(x)^6 * Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^(4*n).
%F A355864 (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.
%e A355864 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 + ...
%e A355864 such that
%e A355864 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) + ...
%e A355864 also
%e A355864 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) * ...
%e A355864 where C(x) = 1 + x*C(x)^2 begins
%e A355864 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 A355864 (PARI) {a(n) = my(A, C=1/x*serreverse(x-x^2 +O(x^(n+2))), M=ceil(sqrt(2*n+9)));
%o A355864 A = sum(m=-M, M, x^(m*(m+1)/2) * C^(4*m-6) ); polcoeff(A, n)}
%o A355864 for(n=0, 70, print1(a(n), ", "))
%Y A355864 Cf. A355341, A355345, A355348, A000108.
%K A355864 sign
%O A355864 0,1
%A A355864 _Paul D. Hanna_, Aug 02 2022