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).

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

Author

Paul D. Hanna, Jul 25 2022

Keywords

• sign

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

• Paul D. Hanna, Table of n, a(n) for n = 0..2555

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.