OFFSET
0,4
COMMENTS
Let C(x) = 1 + x*C(x)^2 be the g.f. of the Catalan numbers (A000108), then
(1) [x^(2*n)] 1/C(-x)^(3*n) = 0 for n >= 1, and
(2) [x^(2*n+1)] 1/C(-x)^(3*n+1) = [x^(2*n+1)] 1/C(-x)^(3*n+2) = 0 for n >= 1.
The g.f. A(x) of this sequence satisfies similar conditions, with the surprisingly simple formula [x^(2*n+1)] A(x)^(3*n+1) = (-1)^n * (3*n+2) * Product_{k=1..n} (3*k + 1) for n >= 1, a conjecture which has been verified for the initial 1001 terms. Compare with A376225, the dual of this sequence.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..502
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^(2*n)] A(x)^(3*n) = 3 for n >= 1.
(2) [x^(2*n+1)] A(x)^(3*n+1) = [x^(2*n+1)] A(x)^(3*n+2) for n >= 1.
(3) [x^(2*n+1)] A(x)^(3*n+1) = (-1)^n * (3*n+2) * Product_{k=1..n} (3*k + 1) for n >= 1 (conjecture).
EXAMPLE
G.f.: A(x) = 1 + x - 6*x^3 + 28*x^4 - 49*x^5 - 209*x^6 + 1945*x^7 - 6300*x^8 - 4446*x^9 + 151432*x^10 - 743536*x^11 + 988060*x^12 + ...
RELATED TABLES.
The table of coefficients of x^k in A(x)^n begins
n\k 0 1 2 3 4 5 6 7 8
1: [1, 1, 0, -6, 28, -49, -209, 1945, -6300, ...];
2: [1, 2, 1, -12, 44, -42, -480, 3136, -7338, ...];
3: [1, 3, 3, -17, 48, 3, -729, 3534, -4749, ...];
4: [1, 4, 6, -20, 41, 68, -896, 3212, -148, ...];
5: [1, 5, 10, -20, 25, 136, -945, 2325, 5010, ...];
6: [1, 6, 15, -16, 3, 192, -863, 1080, 9534, ...];
7: [1, 7, 21, -7, -21, 224, -658, -293, 12565, ...];
8: [1, 8, 28, 8, -42, 224, -356, -1568, 13609, ...];
9: [1, 9, 36, 30, -54, 189, 3, -2547, 12537, ...];
10: [1, 10, 45, 60, -50, 122, 370, -3080, 9555, ...];
11: [1, 11, 55, 99, -22, 33, 693, -3080 5148, ...];
12: [1, 12, 66, 148, 39, -60, 924, -2532, 3, ...];
13: [1, 13, 78, 208, 143, -130, 1027, -1495, -5083, 50960, ...];
14: [1, 14, 91, 280, 301, -140, 987, -96, -9303, 50960, ...];
15: [1, 15, 105, 365, 525, -42, 820, 1485, -11940, 42895, 3, ...]; ...
in which we see that [x^(2*n)] A(x)^(3*n) = 3 for n >= 1.
Also, [x^(2*n+1)] A(x)^(3*n+1) = [x^(2*n+1)] A(x)^(3*n+2) for n >= 1;
these coefficients begin
[-20, 224, -3080, 50960, -990080, 22131200, -559919360, 15823808000, ...],
and appear to equal (-1)^n * (3*n+2) * Product_{k=1..n} (3*k + 1) for n >= 1.
PROG
(PARI) {a(n) = my(V=[1, 1, 0, 0], A); for(i=0, n, V=concat(V, 0); A = Ser(V); m=(#V-1)\2-1;
V[#V-2] = if(#V%2 == 1, -(polcoef(A^(3*m), 2*m) - 3)/(3*m), polcoef(A^(3*m+1), 2*m+1) - polcoef(A^(3*m+2), 2*m+1) ) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 27 2024
STATUS
approved