OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..301
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 34*x^3 + 266*x^4 + 2259*x^5 + 20280*x^6 + 189695*x^7 + 1832274*x^8 + 18164034*x^9 + 183977703*x^10 + ...
where 1 - x = 1 - x*A(x) + x^2*A(x)^4 + x^3*A(x)^9 - x^4*A(x)^16 + x^5*A(x)^25 + x^6*A(x)^36 - x^7*A(x)^49 + x^8*A(x)^64 +-+ ...
The coefficients of x^k in (-1)^(n mod 3)*x^n*A(x)^(n^2) begin:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [0, -1, -1, -5, -34, -266, -2259, -20280, -189695, ...];
n=2: [0, 0, 1, 4, 26, 200, 1683, 14996, 139182, ...];
n=3: [0, 0, 0, 1, 9, 81, 750, 7128, 69237, ...];
n=4: [0, 0, 0, 0, -1, -16, -200, -2304, -25636, ...];
n=5: [0, 0, 0, 0, 0, 1, 25, 425, 6150, ...];
n=6: [0, 0, 0, 0, 0, 0, 1, 36, 810, ...];
n=7: [0, 0, 0, 0, 0, 0, 0, -1, -49, ...];
n=8: [0, 0, 0, 0, 0, 0, 0, 0, 1, ...];
...
the column sums of which equal [1, -1, 0, 0, 0, 0, 0, ...].
PROG
(PARI) {a(n) = my(V=[1, 1], A=1+x); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=0, #A, (-1)^(m%3) * x^m * A^(m^2) ), #V) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 28 2025
STATUS
approved