A355346 G.f.: A(x,y) = Sum_{n=-oo..+oo} (x*y)^(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).

2, -4, 2, -1, -4, 0, -3, 7, 0, 2, -8, 5, 0, -4, 0, -23, 14, 0, 23, 0, 0, -70, 41, 0, 21, 0, 0, 2, -222, 127, 0, 90, 0, 0, -4, 0, -726, 409, 0, 297, 0, 0, 47, 0, 0, -2431, 1355, 0, 1001, 0, 0, 45, 0, 0, 0, -8294, 4587, 0, 3431, 0, 0, 284, 0, 0, 0, 2, -28730, 15795, 0, 11927, 0, 0, 1001, 0, 0, 0, -4, 0, -100776, 55146, 0, 41955, 0, 0, 3640, 0, 0, 0, 79, 0, 0, -357238, 194752, 0, 149072, 0, 0, 13260, 0, 0, 0, 77, 0, 0, 0

Offset: 0

Author

Paul D. Hanna, Jul 25 2022

Keywords

• sign
• tabl

Examples

			G.f.: A(x,y) = 2 + (2*y - 4)*x + (-4*y - 1)*x^2 + (2*y^3 + 7*y - 3)*x^3 + (-4*y^3 + 5*y - 8)*x^4 + (23*y^3 + 14*y - 23)*x^5 + (2*y^6 + 21*y^3 + 41*y - 70)*x^6 + (-4*y^6 + 90*y^3 + 127*y - 222)*x^7 + (47*y^6 + 297*y^3 + 409*y - 726)*x^8 + (45*y^6 + 1001*y^3 + 1355*y - 2431)*x^9 + (2*y^10 + 284*y^6 + 3431*y^3 + 4587*y - 8294)*x^10 + ...
such that
A(x,y) = ... + (x*y)^6/C(x)^9 + (x*y)^3/C(x)^7 + (x*y)/C(x)^5 + 1/C(x)^3 + 1/C(x) + (x*y)*C(x) + (x*y)^3*C(x)^3 + (x*y)^6*C(x)^5 + (x*y)^10*C(x)^7 + (x*y)^15*C(x)^9 + ... + (x*y)^(n*(n+1)/2) * C(x)^(2*n-1) + ...
also
A(x,y) = 1/C(x)^3 * (1 + C(x)^2)*(1 + x*y/C(x)^2)*(1-x) * (1 + x*y*C(x)^2)*(1 + (x*y)^2/C(x)^2)*(1-x^2) * (1 + (x*y)^2*C(x)^2)*(1 + (x*y)^3/C(x)^2)*(1-(x*y)^3) * (1 + (x*y)^3*C(x)^2)*(1 + (x*y)^4/C(x)^2)*(1-(x*y)^4) * ... * (1 + (x*y)^(n-1)*C(x)^2)*(1 + (x*y)^n/C(x)^2)*(1-(x*y)^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 + ...
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, n >= 0, begins:
         2;
        -4,       2;
        -1,      -4, 0;
        -3,       7, 0,       2;
        -8,       5, 0,      -4, 0;
       -23,      14, 0,      23, 0, 0;
       -70,      41, 0,      21, 0, 0,      2;
      -222,     127, 0,      90, 0, 0,     -4, 0;
      -726,     409, 0,     297, 0, 0,     47, 0, 0;
     -2431,    1355, 0,    1001, 0, 0,     45, 0, 0, 0;
     -8294,    4587, 0,    3431, 0, 0,    284, 0, 0, 0,    2;
    -28730,   15795, 0,   11927, 0, 0,   1001, 0, 0, 0,   -4, 0;
   -100776,   55146, 0,   41955, 0, 0,   3640, 0, 0, 0,   79, 0, 0;
   -357238,  194752, 0,  149072, 0, 0,  13260, 0, 0, 0,   77, 0, 0, 0;
  -1277788,  694450, 0,  534251, 0, 0,  48450, 0, 0, 0,  692, 0, 0, 0, 0;
  -4605980, 2496790, 0, 1928992, 0, 0, 177649, 0, 0, 0, 2537, 0, 0, 0, 0, 2;
  ...
the row sums of which yield A355345:
[2, -2, -5, 6, -7, 14, -6, -9, 27, -30, 10, -11, 44, -77, 55, -10, -13, 65, -156, 182, -91, ...].
The row sums in turn form the antidiagonals of the rectangular table given by:
n = 0: [  2,  -5,  14,   -30,    55,    -91,    140, ...];
n = 1: [ -2,  -7,  27,   -77,   182,   -378,    714, ...];
n = 2: [  6,  -9,  44,  -156,   450,  -1122,   2508, ...];
n = 3: [ -6, -11,  65,  -275,   935,  -2717,   7007, ...];
n = 4: [ 10, -13,  90,  -442,  1729,  -5733,  16744, ...];
n = 5: [-10, -15, 119,  -665,  2940, -10948,  35700, ...];
n = 6: [ 14, -17, 152,  -952,  4692, -19380,  69768, ...];
n = 7: [-14, -19, 189, -1311,  7125, -32319, 127281, ...];
...
in which row n has g.f.: (-1)^n*(2*n+1) + (1-x)/(1+x)^(2*n+4) for n >= 0.
		

Crossrefs

Cf. A355345, A000108.

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=0,n+2, (x*y)^(m*(m+1)/2) * (C^(2*m-1) + 1/C^(2*m+3))); 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 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,y) = Sum_{n=-oo..+oo} (x*y)^(n*(n+1)/2) * C(x)^(2*n-1).

(2) A(x,y) = Sum_{n>=0} (x*y)^(n*(n+1)/2) * (C(x)^(2*n-1) + 1/C(x)^(2*n+3)).

(3) A(x,y) = 1/C(x)^3 * Product_{n>=1} (1 + (x*y)^(n-1)*C(x)^2) * (1 + (x*y)^n/C(x)^2) * (1-(x*y)^n), by the Jacobi triple product identity.