cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A355341 G.f.: A(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

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, -9, 27, -30, 9, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, 1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0, 1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0, 1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2022

Keywords

Examples

			G.f.: A(x) = 2 + x - 2*x^2 + x^3 - 3*x^4 + x^6 - 4*x^7 + 2*x^8 + x^10 - 5*x^11 + 5*x^12 + x^15 - 6*x^16 + 9*x^17 - 2*x^18 + x^21 - 7*x^22 + 14*x^23 - 7*x^24 + x^28 - 8*x^29 + 20*x^30 - 16*x^31 + 2*x^32 + x^36 - 9*x^37 + 27*x^38 - 30*x^39 + 9*x^40 + x^45 - 10*x^46 + 35*x^47 - 50*x^48 + 25*x^49 - 2*x^50 + ...
such that
A(x) = ... + x^6/C(x)^4 + x^3/C(x)^3 + x/C(x)^2 + 1/C(x) + 1 + x*C(x) + x^3*C(x)^2 + x^6*C(x)^3 + x^10*C(x)^4 + ...
where
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 + 16796*x^10 + ... + A000108(n)*x^n + ...
The coefficients of x^k in x^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)) begin:
n = 0: [2, -1, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, ...];
n = 1: [0,  2, -1,  1,  3,   9,  28,   90,  297,  1001,  3432,  11934, ...];
n = 2: [0,  0,  0,  2, -1,   5,  13,   39,  123,   401,  1340,   4565, ...];
n = 3: [0,  0,  0,  0,  0,   0,   2,   -1,   11,    28,    89,    293, ...];
n = 4: [0,  0,  0,  0,  0,   0,   0,    0,    0,     0,     2,     -1, ...]; ...
forming a table the column sums of which yield this sequence.
The g.f. may also be written as
A(x) = 2 + (-2*x + 1)*x + (-3*x + 1)*x^3 + (2*x^2 - 4*x + 1)*x^6 + (5*x^2 - 5*x + 1)*x^10 + (-2*x^3 + 9*x^2 - 6*x + 1)*x^15 + (-7*x^3 + 14*x^2 - 7*x + 1)*x^21 + (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)*x^28 + (9*x^4 - 30*x^3 + 27*x^2 - 9*x + 1)*x^36 + (-2*x^5 + 25*x^4 - 50*x^3 + 35*x^2 - 10*x + 1)*x^45 + ...
compare to
(1 - 2*y*x)/(1-x + y*x^2) = 1 + (-2*y + 1)*x + (-3*y + 1)*x^2 + (2*y^2 - 4*y + 1)*x^3 + (5*y^2 - 5*y + 1)*x^4 + (-2*y^3 + 9*y^2 - 6*y + 1)*x^5 + (-7*y^3 + 14*y^2 - 7*y + 1)*x^6 + (2*y^4 - 16*y^3 + 20*y^2 - 8*y + 1)*x^7 + (9*y^4 - 30*y^3 + 27*y^2 - 9*y + 1)*x^8 + (-2*y^5 + 25*y^4 - 50*y^3 + 35*y^2 - 10*y + 1)*x^9 + ...
The terms of this sequence may be written as a triangle (see triangle A244422):
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, -9, 27, -30, 9, 0, 0, 0, 0,
1, -10, 35, -50, 25, -2, 0, 0, 0, 0,
1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0,
1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0,
1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0,
1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0,
1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0,
1, -16, 104, -352, 660, -672, 336, -64, 2, 0, 0, 0, 0, 0, 0, 0,
...

Links

Crossrefs

Cf. A244422, A355342, A355343.

Programs

  • PARI
    {a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x);
A = sum(m=-n-1,n+1, x^(m*(m+1)/2) * C^m); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))
  • PARI
    {a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
A = sum(m=0,M, x^(m*(m+1)/2) * (C^m + 1/C^(m+1))); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n is equal to the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(1) A(x) = 1/C(x) * Product_{n>=1} (1 + x^n/C(x)) * (1 + x^(n-1)*C(x)) * (1-x^n), by the Jacobi triple product identity.
(2) A(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^n.
(3) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)).
(4) A(x) = 1 + Sum_{n>=0} x^(n*(n+1)/2) * ( [y^n] (1 - 2*y*x)/(1-y + x*y^2) ).
(5) A(x) = 1 + Sum_{n>=1} x^(n*(n-1)/2) * Sum_{k=0..n} A244422(n,k) * x^k.

A355345 G.f.: Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^(2*n-1), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

2, -2, -5, 6, -7, 14, -6, -9, 27, -30, 10, -11, 44, -77, 55, -10, -13, 65, -156, 182, -91, 14, -15, 90, -275, 450, -378, 140, -14, -17, 119, -442, 935, -1122, 714, -204, 18, -19, 152, -665, 1729, -2717, 2508, -1254, 285, -18, -21, 189, -952, 2940, -5733, 7007, -5148, 2079, -385, 22, -23, 230, -1311, 4692, -10948, 16744, -16445, 9867, -3289, 506
Offset: 0

Views

Author

Paul D. Hanna, Jul 25 2022

Keywords

Examples

			G.f.: A(x) = 2 - 2*x - 5*x^2 + 6*x^3 - 7*x^4 + 14*x^5 - 6*x^6 - 9*x^7 + 27*x^8 - 30*x^9 + 10*x^10 - 11*x^11 + 44*x^12 - 77*x^13 + 55*x^14 - 10*x^15 - 13*x^16 + 65*x^17 - 156*x^18 + 182*x^19 - 91*x^20 + ...
such that
A(x) = ... + x^6/C(x)^9 + x^3/C(x)^7 + x/C(x)^5 + 1/C(x)^3 + 1/C(x) + x*C(x) + x^3*C(x)^3 + x^6*C(x)^5 + x^10*C(x)^7 + x^15*C(x)^9 + ... + x^(n*(n+1)/2) * C(x)^(2*n-1) + ...
also
A(x) = 1/C(x)^3 * (1 + C(x)^2)*(1 + x/C(x)^2)*(1-x) * (1 + x*C(x)^2)*(1 + x^2/C(x)^2)*(1-x^2) * (1 + x^2*C(x)^2)*(1 + x^3/C(x)^2)*(1-x^3) * (1 + x^3*C(x)^2)*(1 + x^4/C(x)^2)*(1-x^4) * ... * (1 + x^(n-1)*C(x)^2)*(1 + x^n/C(x)^2)*(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 + ...
RELATED TABLE.
This sequence also forms the antidiagonals of the rectangular table given by:
n = 0: [  2,  -5,  14,   -30,    55,    -91,    140,    -204, ...];
n = 1: [ -2,  -7,  27,   -77,   182,   -378,    714,   -1254, ...];
n = 2: [  6,  -9,  44,  -156,   450,  -1122,   2508,   -5148, ...];
n = 3: [ -6, -11,  65,  -275,   935,  -2717,   7007,  -16445, ...];
n = 4: [ 10, -13,  90,  -442,  1729,  -5733,  16744,  -44200, ...];
n = 5: [-10, -15, 119,  -665,  2940, -10948,  35700, -104652, ...];
n = 6: [ 14, -17, 152,  -952,  4692, -19380,  69768, -224808, ...];
n = 7: [-14, -19, 189, -1311,  7125, -32319, 127281, -447051, ...];
n = 8: [ 18, -21, 230, -1750, 10395, -51359, 219604, -834900, ...];
...
in which row n has g.f.: (-1)^n*(2*n+1) + (1-x)/(1+x)^(2*n+4) for n >= 0.
Thus, the terms of this sequence obey the rule
a((n+k)*(n+k+1)/2 + k) = [x^k] ((-1)^n*(2*n+1) + (1-x)/(1+x)^(2*n+4)), for n >= 0, k = 0..n.
Equivalently,
a((n+k)*(n+k+1)/2 + k) = (-1)^k*(binomial(2*n+k+3,k) + binomial(2*n+k+2,k-1)), for n >= 0, k >= 1, with a(n*(2*n+1)) = 2*(2*n+1) and a((n+1)*(2*n+1)) = -2*(2*n+1) for n >= 0.
For example,
a((n+1)*(n+2)/2 + 1) = -(2*n+5) for n >= 0,
a((n+2)*(n+3)/2 + 2) = (n+2)*(2*n+7) for n >= 0,
a(n*(n+3)/2) = (-1)^n * (n+1)*(n+2)*(2*n+3)/6  for n >= 1,
a(2*n*(n+1)) = (-1)^n * (binomial(3*n+3,n) + binomial(3*n+2,n-1)) = (-1)^n * A355347(n), for n >= 1.
...

Links

Crossrefs

Cf. A132460, A034807, A000108, A355341, A355342, A355346, A355347.

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^(2*m-1) ); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))
  • PARI
    {a(n) = my(A,M=ceil(sqrt(2*n+1)));
A = sum(m=0,M, sum(k=0,n-m*(m+1)/2, x^((m+k)*(m+k+1)/2 + k) * polcoeff( (-1)^m*(2*m+1) + (1-x)/(1+x +x^2*O(x^k))^(2*m+4) ,k) )); 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)^(2*n-1).
(2) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * (C(x)^(2*n-1) + 1/C(x)^(2*n+3)).
(3) A(x) = 1/C(x)^3 * Product_{n>=1} (1 + x^(n-1)*C(x)^2) * (1 + x^n/C(x)^2) * (1-x^n), by the Jacobi triple product identity.
(4) A(x) = 1/P(x)^3 + Sum_{n>=0} Sum_{k>=0} (-1)^k * (binomial(2*n+k+3,k) + binomial(2*n+k+2,k-1)) * x^((n+k)*(n+k+1)/2 + k), where P(x) = Product_{n>=1} 1/(1-x^n) is the partition function.
(5) a((n+k)*(n+k+1)/2 + k) = [x^k] (-1)^n*(2*n+1) + (1-x)/(1+x)^(2*n+4), for n >= 0, k >= 0.
(6) a((n+k)*(n+k+1)/2 + k) = (-1)^k*(binomial(2*n+k+3,k) + binomial(2*n+k+2,k-1)), for n >= 0, k >= 1.

A356777 G.f.: Sum_{n=-oo..+oo} x^(n^2) * C(x)^(2*n-1), where C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, -3, 0, 1, -5, 5, 0, 0, 1, -7, 14, -7, 0, 0, 0, 1, -9, 27, -30, 9, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0, 1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0, 1, -17, 119, -442, 935, -1122, 714, -204, 17, 0, 0, 0, 0, 0, 0, 0, 0, 1, -19, 152, -665, 1729, -2717, 2508, -1254, 285, -19
Offset: 0

Views

Author

Paul D. Hanna, Sep 08 2022

Keywords

Examples

			G.f.: A(x) = 1 + x - 3*x^2 + x^4 - 5*x^5 + 5*x^6 + x^9 - 7*x^10 + 14*x^11 - 7*x^12 + x^16 - 9*x^17 + 27*x^18 - 30*x^19 + 9*x^20 + x^25 - 11*x^26 + 44*x^27 - 77*x^28 + 55*x^29 - 11*x^30 + x^36 - 13*x^37 + 65*x^38 - 156*x^39 + 182*x^40 - 91*x^41 + 13*x^42 + x^49 - 15*x^50 + 90*x^51 - 275*x^52 + 450*x^53 - 378*x^54 + 140*x^55 - 15*x^56 + ...
such that
A(x) = ... + x^16/C(x)^9 + x^9/C(x)^7 + x^4/C(x)^5 + x/C(x)^3 + 1/C(x) + x*C(x) + x^4*C(x)^3 + x^9*C(x)^5 + x^16*C(x)^7 + x^25*C(x)^9 + ... + x^(n^2)*C^(2*n-1) + ...
where the Catalan function C(x) = (1 - sqrt(1-4*x))/(2*x) 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 + 4862*x^9 + ... + A000108(n)*x^n + ...
RELATED TABLE.
This sequence may be written in the form of an irregular triangle:
1,
1, -3, 0,
1, -5, 5, 0, 0,
1, -7, 14, -7, 0, 0, 0,
1, -9, 27, -30, 9, 0, 0, 0, 0,
1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0,
1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0,
1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0,
1, -17, 119, -442, 935, -1122, 714, -204, 17, 0, 0, 0, 0, 0, 0, 0, 0,
1, -19, 152, -665, 1729, -2717, 2508, -1254, 285, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0,
...
Compare the above construction to triangle A082985.

Links

Crossrefs

Cf. A355341, A355342, A355345, A356778, A082985, A000108.

Programs

  • PARI
    /* By Definition: */
{a(n) = my(A, C = 1/x*serreverse(x-x^2 +O(x^(n+2))), M=ceil(sqrt(n+1)));
A = sum(m=-M, M, x^(m^2) * C^(2*m-1) ); polcoeff(A, n)}
for(n=0, 90, print1(a(n), ", "))
  • PARI
    /* Without Using Catalan Series */
{a(n) = my(A, M=ceil(sqrt(n+1)));
A = sum(m=0, M, sum(k=0, 2*m, (-1)^k*binomial(2*m-k, k)*(2*m+1)/(2*m-2*k+1) * x^(m^2 + k) ) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 90, 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^2) * C(x)^(2*n-1).
(2) A(x) = 1/C(x) * Product_{n>=1} (1 + x^(2*n-1)*C(x)^2) * (1 + x^(2*n-1)/C(x)^2) * (1 - x^(2*n)), by the Jacobi triple product identity.
(3) A(x) = 1/C(x) + Sum_{n>=1} x^(n^2) * (C(x)^(2*n-1) + 1/C(x)^(2*n+1)).
(4) A(x) = Sum_{n>=0} Sum_{k=0..n} (-1)^k * binomial(2*n-k, k) * (2*n+1)/(2*n-2*k+1) * x^(n^2 + k).

A355344 G.f.: A(x,y) = Sum_{n=-oo..+oo} (-1)^n * (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.

Original entry on oeis.org

0, 1, 0, 1, -3, 0, 2, -3, 0, 0, 5, -7, 0, 5, 0, 14, -19, 0, 5, 0, 0, 42, -56, 0, 15, 0, 0, 0, 132, -174, 0, 45, 0, 0, -7, 0, 429, -561, 0, 141, 0, 0, -7, 0, 0, 1430, -1859, 0, 457, 0, 0, -28, 0, 0, 0, 4862, -6292, 0, 1520, 0, 0, -91, 0, 0, 0, 0, 16796, -21658, 0, 5159, 0, 0, -301, 0, 0, 0, 9, 0, 58786, -75582, 0, 17797, 0, 0, -1015, 0, 0, 0, 9, 0, 0, 208012
Offset: 0

Views

Author

Paul D. Hanna, Jul 22 2022

Keywords

Examples

			G.f.: A(x,y) = x + (-3*y + 1)*x^2 + (-3*y + 2)*x^3 + (5*y^3 - 7*y + 5)*x^4 + (5*y^3 - 19*y + 14)*x^5 + (15*y^3 - 56*y + 42)*x^6 + (-7*y^6 + 45*y^3 - 174*y + 132)*x^7 + (-7*y^6 + 141*y^3 - 561*y + 429)*x^8 + (-28*y^6 + 457*y^3 - 1859*y + 1430)*x^9 + ...
where
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 +- ... + (-1)^n * (x*y)^(n*(n+1)/2) * C(x)^n + ...
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:
n = 0: [0];
n = 1: [1, 0];
n = 2: [1, -3, 0];
n = 3: [2, -3, 0, 0];
n = 4: [5, -7, 0, 5, 0];
n = 5: [14, -19, 0, 5, 0, 0];
n = 6: [42, -56, 0, 15, 0, 0, 0];
n = 7: [132, -174, 0, 45, 0, 0, -7, 0];
n = 8: [429, -561, 0, 141, 0, 0, -7, 0, 0];
n = 9: [1430, -1859, 0, 457, 0, 0, -28, 0, 0, 0];
n = 10: [4862, -6292, 0, 1520, 0, 0, -91, 0, 0, 0, 0];
n = 11: [16796, -21658, 0, 5159, 0, 0, -301, 0, 0, 0, 9, 0];
n = 12: [58786, -75582, 0, 17797, 0, 0, -1015, 0, 0, 0, 9, 0, 0];
n = 13: [208012, -266798, 0, 62218, 0, 0, -3480, 0, 0, 0, 48, 0, 0, 0];
n = 14: [742900, -950912, 0, 219946, 0, 0, -12099, 0, 0, 0, 165, 0, 0, 0, 0];
n = 15: [2674440, -3417340, 0, 784890, 0, 0, -42562, 0, 0, 0, 573, 0, 0, 0, 0, 0];
n = 16: [9694845, -12369285, 0, 2823666, 0, 0, -151228, 0, 0, 0, 2007, 0, 0, 0, 0, -11, 0];
...
The row sums of this triangle form sequence A355342:
[0, 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, ...],
which in turn may be written in the form of a triangle:
0,
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, 9, -27, 30, -9, 0, 0, 0, 0,
1, -10, 35, -50, 25, -2, 0, 0, 0, 0,
-1, 11, -44, 77, -55, 11, 0, 0, 0, 0, 0,
1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0,
...

Links

Crossrefs

Cf. A000108, A355342, A355343.

Programs

  • PARI
    {T(n,k) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
A = sum(m=-M,M, (-1)^m * (x*y)^(m*(m+1)/2) * C^m); polcoeff(polcoeff(A,n,x),k,y)}
for(n=0,16,for(k=0,n, print1(T(n,k),", "));print(""))
  • PARI
    {T(n,k) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
A = sum(m=0,n+2, (-1)^m * (x*y)^(m*(m+1)/2) * (C^m - 1/C^(m+1))); polcoeff(polcoeff(A,n,x),k,y)}
for(n=0,16,for(k=0,n, print1(T(n,k),", "));print(""))

Formula

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