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%I #6 Jul 24 2022 03:57:36
%S 2,-1,2,-1,-1,0,-2,1,0,2,-5,3,0,-1,0,-14,9,0,5,0,0,-42,28,0,13,0,0,2,
%T -132,90,0,39,0,0,-1,0,-429,297,0,123,0,0,11,0,0,-1430,1001,0,401,0,0,
%U 28,0,0,0,-4862,3432,0,1340,0,0,89,0,0,0,2,-16796,11934,0,4565,0,0,293,0,0,0,-1,0,-58786,41990,0,15795,0,0,987,0,0,0,19,0,0,-208012
%N G.f.: A(x,y) = Sum_{n=-oo..+oo} (x*y)^(n*(n+1)/2) * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.
%H Paul D. Hanna, <a href="/A355343/b355343.txt">Table of n, a(n) for n = 0..1325</a>
%F G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * y^k is equal to the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%F (1) A(x,y) = Sum_{n=-oo..+oo} (x*y)^(n*(n+1)/2) * C(x)^n,
%F (2) A(x,y) = Sum_{n>=0} (x*y)^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)).
%F (3) A(x,y) = 1/C(x) * Product_{n>=1} (1 + (x*y)^n/C(x)) * (1 + (x*y)^(n-1)*C(x)) * (1-(x*y)^n), by the Jacobi triple product identity.
%e G.f.: A(x,y) = 2 + (2*y - 1)*x + (-y - 1)*x^2 + (2*y^3 + y - 2)*x^3 + (-y^3 + 3*y - 5)*x^4 + (5*y^3 + 9*y - 14)*x^5 + (2*y^6 + 13*y^3 + 28*y - 42)*x^6 + (-y^6 + 39*y^3 + 90*y - 132)*x^7 + (11*y^6 + 123*y^3 + 297*y - 429)*x^8 + (28*y^6 + 401*y^3 + 1001*y - 1430)*x^9 + ...
%e where
%e A(x,y) = ... + (x*y)^6/C(x)^4 + (x*y)^3/C(x)^3 + (x*y)/C(x)^2 + 1/C(x) + 1 + (x*y)*C(x) + (x*y)^3*C(x)^2 + (x*y)^6*C(x)^3 + ... + (x*y)^(n*(n+1)/2) * C(x)^n + ...
%e This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y), for k = 0..n in row n, begins:
%e n = 0: [2];
%e n = 1: [-1, 2];
%e n = 2: [-1, -1, 0];
%e n = 3: [-2, 1, 0, 2];
%e n = 4: [-5, 3, 0, -1, 0];
%e n = 5: [-14, 9, 0, 5, 0, 0];
%e n = 6: [-42, 28, 0, 13, 0, 0, 2];
%e n = 7: [-132, 90, 0, 39, 0, 0, -1, 0];
%e n = 8: [-429, 297, 0, 123, 0, 0, 11, 0, 0];
%e n = 9: [-1430, 1001, 0, 401, 0, 0, 28, 0, 0, 0];
%e n = 10: [-4862, 3432, 0, 1340, 0, 0, 89, 0, 0, 0, 2];
%e n = 11: [-16796, 11934, 0, 4565, 0, 0, 293, 0, 0, 0, -1, 0];
%e n = 12: [-58786, 41990, 0, 15795, 0, 0, 987, 0, 0, 0, 19, 0, 0];
%e n = 13: [-208012, 149226, 0, 55354, 0, 0, 3384, 0, 0, 0, 48, 0, 0, 0];
%e n = 14: [-742900, 534888, 0, 196078, 0, 0, 11769, 0, 0, 0, 165, 0, 0, 0, 0];
%e n = 15: [-2674440, 1931540, 0, 700910, 0, 0, 41418, 0, 0, 0, 571, 0, 0, 0, 0, 2];
%e n = 16: [-9694845, 7020405, 0, 2525214, 0, 0, 147224, 0, 0, 0, 1997, 0, 0, 0, 0, -1, 0];
%e ...
%e The row sums of this triangle form sequence A355341:
%e [2, 1, -2, 1, -3, 0, 1, -4, 2, 0, 1, -5, 5, 0, 0, 1, -6, 9, -2, 0, 0, 1, -7, 14, -7, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, 1, ...],
%e which in turn may be written in the form of a triangle:
%e 2,
%e 1, -2,
%e 1, -3, 0,
%e 1, -4, 2, 0,
%e 1, -5, 5, 0, 0,
%e 1, -6, 9, -2, 0, 0,
%e 1, -7, 14, -7, 0, 0, 0,
%e 1, -8, 20, -16, 2, 0, 0, 0,
%e 1, -9, 27, -30, 9, 0, 0, 0, 0,
%e 1, -10, 35, -50, 25, -2, 0, 0, 0, 0,
%e 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0,
%e 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0,
%e ...
%o (PARI) {T(n,k) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9)););
%o A = sum(m=-M,M, (x*y)^(m*(m+1)/2) * C^m); polcoeff(polcoeff(A,n,x),k,y)}
%o for(n=0,16,for(k=0,n, print1(T(n,k),", "));print(""))
%o (PARI) {T(n,k) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
%o A = sum(m=0,n+2, (x*y)^(m*(m+1)/2) * (C^m + 1/C^(m+1))); polcoeff(polcoeff(A,n,x),k,y)}
%o for(n=0,16,for(k=0,n, print1(T(n,k),", "));print(""))
%Y Cf. A000108, A355341, A355344.
%K sign,tabl
%O 0,1
%A _Paul D. Hanna_, Jul 22 2022
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