OFFSET
0,5
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1325
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.
EXAMPLE
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,
...
PROG
(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(""))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Jul 22 2022
STATUS
approved