OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..2555
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)^(6*n-9).
(2) A(x) = 1/C(x)^9 * Product_{n>=1} (1 + x^(2*n-1)*C(x)^6) * (1 + x^(2*n-1)/C(x)^6) * (1 - x^(2*n)), by the Jacobi triple product identity.
(3) A(x) = 1/C(x)^9 + Sum_{n>=1} x^(n^2) * (C(x)^(6*n-9) + 1/C(x)^(6*n+9)).
EXAMPLE
G.f.: A(x) = 1 - 7*x + 9*x^2 + 60*x^3 - 265*x^4 + 429*x^5 - 189*x^6 - 812*x^7 + 2925*x^8 - 5732*x^9 + 6980*x^10 - 4824*x^11 - 198*x^12 + 10010*x^13 - 32298*x^14 + 69768*x^15 - 104651*x^16 + 107373*x^17 - 72435*x^18 + 26422*x^19 + 19656*x^20 - 115011*x^21 + 361763*x^22 - 834900*x^23 + 1427679*x^24 - 1797817*x^25 + ...
such that
A(x) = ... + x^16/C(x)^33 + x^9/C(x)^27 + x^4/C(x)^21 + x/C(x)^15 + 1/C(x)^9 + x/C(x)^3 + x^4*C(x)^3 + x^9*C(x)^9 + x^16*C(x)^15 + x^25*C(x)^21 + ... + x^(n^2)*C(x)^(6*n-9) + ...
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 that begins:
1,
-7, 9, 60,
-265, 429, -189, -812, 2925,
-5732, 6980, -4824, -198, 10010, -32298, 69768,
-104651, 107373, -72435, 26422, 19656, -115011, 361763, -834900, 1427679,
-1797817, 1641447, -1057446, 454155, -69564, -298980, 1307448, -4102104, 9924525, -18599295, 26936910,
-29910464, 25109975, -15599955, 6941244, -2013544, -324558, 3717882, -14942570, 46955661, -117679100, 236030652, -378658800, 483841800,
...
PROG
(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^(6*m-9) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Sep 08 2022
STATUS
approved