login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A355342
G.f.: A(x) = Sum_{n=-oo..+oo} (-1)^n * 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).
5
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, -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
0,3
LINKS
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} (-1)^n * x^(n*(n+1)/2) * C(x)^n.
(3) A(x) = Sum_{n>=0} (-1)^n * 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+x + y*x^2) ).
(5) A(x) = 1 - Sum_{n>=1} (-1)^n * x^(n*(n-1)/2) * Sum_{k=0..n} A244422(n,k) * x^k.
EXAMPLE
G.f.: A(x) = 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 (-1)^n * x^(n*(n+1)/2) * (C(x)^n - 1/C(x)^(n+1)) begin:
n = 0: [0, 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ...];
n = 1: [0, 0, -3, -3, -7, -19, -56, -174, -561, -1859, -6292, -21658, ...];
n = 2: [0, 0, 0, 0, 5, 5, 15, 45, 141, 457, 1520, 5159, ...];
n = 3: [0, 0, 0, 0, 0, 0, 0, -7, -7, -28, -91, -301, ...];
n = 4: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, ...]; ...
forming a table the column sums of which yield this sequence.
The g.f. may also be written as
A(x) = 0 + (-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:
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,
-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,
...
PROG
(PARI) {a(n) = my(A, C = serreverse(x-x^2 +x^2*O(x^n))/x);
A = sum(m=-n-1, n+1, (-1)^m * 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, (-1)^m * x^(m*(m+1)/2) * (C^m - 1/C^(m+1))); polcoeff(A, n)}
for(n=0, 70, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 22 2022
STATUS
approved