OFFSET
0,4
COMMENTS
It appears that [x^(n-1)] A'(x)/A(x) is divisible by prime(n) for n > 1 (see example section).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
EXAMPLE
G.f.: A(x) = 1 + x - x^2 + 2*x^3 - 5*x^4 + 13*x^5 - 31*x^6 + 48*x^7 + 129*x^8 - 2035*x^9 + 12963*x^10 - 20703*x^11 - 782282*x^12 + 14675113*x^13 - 177056253*x^14 + 1716591959*x^15 + ...
We illustrate the defining property of this sequence below.
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, -1, 2, -5, 13, -31, 48, 129, ...];
2: [1, 2, -1, 2, -5, 12, -22, -12, 493, ...];
3: [1, 3, (0), 1, -3, 6, -1, -93, 834, ...];
4: [1, 4, 2, 0, -1, 0, 18, -156, 1055, ...];
5: [1, 5, 5, (0), 0, -4, 30, -190, 1145, ...];
6: [1, 6, 9, 2, 0, -6, 35, -198, 1131, ...];
7: [1, 7, 14, 7, (0), -7, 35, -188, 1050, ...];
8: [1, 8, 20, 16, 2, -8, 32, -168, 935, ...];
9: [1, 9, 27, 30, 9, -9, 27, -144, 810, ...];
10: [1, 10, 35, 50, 25, -8, 20, -120, 690, ...];
11: [1, 11, 44, 77, 55, (0), 11, -99, 583, ...];
12: [1, 12, 54, 112, 105, 24, 2, -84, 492, ...];
13: [1, 13, 65, 156, 182, 78, (0), -78, 416, ...];
14: [1, 14, 77, 210, 294, 182, 21, -82, 350, ...];
15: [1, 15, 90, 275, 450, 363, 95, -90, 285, ...];
16: [1, 16, 104, 352, 660, 656, 272, -80, 210, ...];
17: [1, 17, 119, 442, 935, 1105, 629, (0), 119, ...];
18: [1, 18, 135, 546, 1287, 1764, 1278, 252, 27, ...];
19: [1, 19, 152, 665, 1729, 2698, 2375, 855, (0), ...];
...
in which [x^n] A(x)^prime(n) = 0 for n > 1.
RELATED SEQUENCES.
The logarithmic derivative A'(x)/A(x) yields sequence A381355
L = [1, -3, 10, -35, 121, -390, 1037, -1083, -14030, 137837, -382106, -8791718, 199408912, -2701500413, 28888970650, ..., A381355(n), ...]
where L(n)/prime(n) for n > 1 appears to form an integer sequence
[-1, 2, -5, 11, -30, 61, -57, -610, 4753, -12326, -237614, 4863632, -62825591, 614658950, -4949571887, 35267329190, -227575820550, ...].
The sequence defined by [x^(n-1)] A(x)^prime(n)/prime(n) for n > 1 begins
[1, 1, 1, 5, 6, 37, 45, 316, 7096, 9670, 224327, 1236047, 1671349, 11090210, 243407086, 4100579442, 5929250052, 128190709101, 709594806050, 970528475660, 23882286084839, 135758592004780, 2441501831566717, 97016704435777134, ...]
as can be seen from the above table.
PROG
(PARI) {a(n) = my(A=[1, 1]); for(m=2, n+1, A=concat(A, 0);
A[#A] = -polcoef(Ser(A)^prime(m)/prime(m), m) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 11 2025
STATUS
approved