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A356778 G.f.: Sum_{n=-oo..+oo} x^(n^2) * C(x)^(4*n-4), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

%I A356778 #12 Sep 13 2022 04:43:13
%S A356778 1,-2,-6,20,-15,-10,54,-112,105,-35,-14,104,-352,660,-672,336,-63,-18,
%T A356778 170,-800,2275,-4004,4290,-2640,825,-99,-22,252,-1520,5814,-14688,
%U A356778 24752,-27456,19305,-8008,1716,-143,-26,350,-2576,12397,-40964,94962,-155040,176358,-136136,68068,-20384,3185
%N A356778 G.f.: Sum_{n=-oo..+oo} x^(n^2) * C(x)^(4*n-4), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%H A356778 Paul D. Hanna, <a href="/A356778/b356778.txt">Table of n, a(n) for n = 0..2555</a>
%F A356778 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 A356778 (1) A(x) = Sum_{n=-oo..+oo} x^(n^2) * C(x)^(4*n-4).
%F A356778 (2) A(x) = 1/C(x)^4 * Product_{n>=1} (1 + x^(2*n-1)*C(x)^4) * (1 + x^(2*n-1)/C(x)^4) * (1 - x^(2*n)), by the Jacobi triple product identity.
%F A356778 (3) A(x) = 1/C(x)^4 + Sum_{n>=1} x^(n^2) * (C(x)^(4*n-4) + 1/C(x)^(4*n+4)).
%F A356778 (4) A(x) = Sum_{n>=0} Sum_{k=0..2*n} (-1)^k * (binomial(4*n-k,k) + binomial(4*n-k-1,k-1)) * x^((n-1)^2 + k).
%e A356778 G.f.: A(x) = 1 - 2*x - 6*x^2 + 20*x^3 - 15*x^4 - 10*x^5 + 54*x^6 - 112*x^7 + 105*x^8 - 35*x^9 - 14*x^10 + 104*x^11 - 352*x^12 + 660*x^13 - 672*x^14 + 336*x^15 - 63*x^16 - 18*x^17 + 170*x^18 - 800*x^19 + 2275*x^20 - 4004*x^21 + 4290*x^22 - 2640*x^23 + 825*x^24 - 99*x^25 - 22*x^26 + 252*x^27 - 1520*x^28 + 5814*x^29 - 14688*x^30 + 24752*x^31 - 27456*x^32 + 19305*x^33 - 8008*x^34 + 1716*x^35 - 143*x^36 + ...
%e A356778 such that
%e A356778 A(x) = ... + x^16/C(x)^20 + x^9/C(x)^16 + x^4/C(x)^12 + x/C(x)^8 + 1/C(x)^4 + x + x^4*C(x)^4 + x^9*C(x)^8 + x^16*C(x)^12 + x^25*C(x)^16 + ... + x^(n^2)*C(x)^(4*n-4) + ...
%e A356778 where the Catalan function C(x) = (1 - sqrt(1-4*x))/(2*x) begins
%e A356778 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 + 4862*x^9 + ... + A000108(n)*x^n + ...
%e A356778 RELATED TABLE.
%e A356778 This sequence may be written in the form of an irregular triangle that begins:
%e A356778 1,
%e A356778 -2, -6, 20,
%e A356778 -15, -10, 54, -112, 105,
%e A356778 -35, -14, 104, -352, 660, -672, 336,
%e A356778 -63, -18, 170, -800, 2275, -4004, 4290, -2640, 825,
%e A356778 -99, -22, 252, -1520, 5814, -14688, 24752, -27456, 19305, -8008, 1716,
%e A356778 -143, -26, 350, -2576, 12397, -40964, 94962, -155040, 176358, -136136, 68068, -20384, 3185,
%e A356778 -195, -30, 464, -4032, 23400, -95680, 283360, -615296, 980628, -1136960, 940576, -537472, 201552, -45696, 5440,
%e A356778 ...
%e A356778 Compare the above to a related table B where B(n,k) = (-1)^k * A034807(4*n,k), for n >= 0, k = 0.. 2*n, and starts as:
%e A356778 1,
%e A356778 1, -4, 2,
%e A356778 1, -8, 20, -16, 2,
%e A356778 1, -12, 54, -112, 105, -36, 2,
%e A356778 1, -16, 104, -352, 660, -672, 336, -64, 2,
%e A356778 1, -20, 170, -800, 2275, -4004, 4290, -2640, 825, -100, 2,
%e A356778 1, -24, 252, -1520, 5814, -14688, 24752, -27456, 19305, -8008, 1716, -144, 2,
%e A356778 ...
%o A356778 (PARI) /* By Definition: */
%o A356778 {a(n) = my(A, C=1/x*serreverse(x-x^2 +O(x^(n+2))), M=ceil(sqrt(n+1)));
%o A356778 A = sum(m=-M, M, x^(m^2) * C^(4*m-4) ); polcoeff(A, n)}
%o A356778 for(n=0, 50, print1(a(n), ", "))
%o A356778 (PARI) /* Without Using Catalan Series - Faster */
%o A356778 {a(n) = my(A, M=ceil(sqrt(n+1)));
%o A356778 A = sum(m=0, M, sum(k=0, 2*m, (-1)^k * (binomial(4*m-k,k) + binomial(4*m-k-1,k-1)) * x^((m-1)^2 + k) ) +x*O(x^n)); polcoeff(A, n)}
%o A356778 for(n=0, 50, print1(a(n), ", "))
%Y A356778 Cf. A355341, A355345, A034807, A356777, A000108.
%K A356778 sign
%O A356778 0,2
%A A356778 _Paul D. Hanna_, Sep 08 2022