OFFSET
0,3
COMMENTS
Let D(x) = 1 + x*D(x)^3 be the g.f. of A001764, then
(1) [x^(2*n)] 1/D(-x)^(5*n) = 0 for n >= 1, and
(2) [x^(2*n+1)] 1/D(-x)^(5*n+2) = [x^(2*n+1)] 1/D(-x)^(5*n+3) = 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)^(5*n+2) = (5*n+2) * Product_{k=1..n} (5*k + 3) for n >= 1, a conjecture which has been verified for the initial 1001 terms. Compare with A377097, the dual of this sequence.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..600
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^(2*n)] A(x)^(5*n) = -5 for n >= 1.
(2) [x^(2*n+1)] A(x)^(5*n+2) = [x^(2*n+1)] A(x)^(5*n+3) for n >= 1.
(3) [x^(2*n+1)] A(x)^(5*n+2) = (5*n+2) * Product_{k=1..n} (5*k + 3) for n >= 1 (conjecture).
EXAMPLE
G.f.: A(x) = 1 + x - 3*x^2 + 21*x^3 - 143*x^4 + 1149*x^5 - 9770*x^6 + 86972*x^7 - 800114*x^8 + 7550103*x^9 - 72668622*x^10 + 710843094*x^11 - 7046801413*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
1: [1, 1, -3, 21, -143, 1149, -9770, 86972, ...];
2: [1, 2, -5, 36, -235, 1886, -15943, 141504, ...];
3: [1, 3, -6, 46, -285, 2301, -19353, 171507, ...];
4: [1, 4, -6, 52, -301, 2472, -20696, 183456, ...];
5: [1, 5, -5, 55, -290, 2466, -20545, 182585, ...];
6: [1, 6, -3, 56, -258, 2340, -19364, 173064, ...];
7: [1, 7, 0, 56, -210, 2142, -17521, 158159, ...];
8: [1, 8, 4, 56, -150, 1912, -15300, 140376, ...];
9: [1, 9, 9, 57, -81, 1683, -12912, 121590, ...];
10: [1, 10, 15, 60, -5, 1482, -10505, 103160, ...];
11: [1, 11, 22, 66, 77, 1331, -8173, 86031, ...];
12: [1, 12, 30, 76, 165, 1248, -5964, 70824, ...];
13: [1, 13, 39, 91, 260, 1248, -3887, 57915, ...];
14: [1, 14, 49, 112, 364, 1344, -1918, 47504, ...];
15: [1, 15, 60, 140, 480, 1548, -5, 39675, ...];
16: [1, 16, 72, 176, 612, 1872, 1928, 34448, ...];
17: [1, 17, 85, 221, 765, 2329, 3978, 31824, ...];
18: [1, 18, 99, 276, 945, 2934, 6261, 31824, ...];
19: [1, 19, 114, 342, 1159, 3705, 8911, 34523, ...];
20: [1, 20, 130, 420, 1415, 4664, 12080, 40080, -5, ...];
21: [1, 21, 147, 511, 1722, 5838, 15939, 48765, 51240, 1023701, ...];
22: [1, 22, 165, 616, 2090, 7260, 20680, 60984, 105787, 947232, ...];
23: [1, 23, 184, 736, 2530, 8970, 26519, 77303, 167325, 947232, ...];
...
in which we see that [x^(2*n)] A(x)^(5*n) = -5 for n >= 1.
Also, [x^(2*n+1)] A(x)^(5*n+2) = [x^(2*n+1)] A(x)^(5*n+3) for n >= 1;
these coefficients begin
[56, 1248, 31824, 947232, 32550336, 1273079808, ...],
and appear to equal (5*n + 2) * Product_{k=1..n} (5*k + 3) 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^(5*m), 2*m) + 5)/(5*m), polcoef(A^(5*m+2), 2*m+1) - polcoef(A^(5*m+3), 2*m+1) ) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 04 2024
STATUS
approved