OFFSET
0,4
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+3) * Product_{k=1..n} (5*k + 2) for n >= 1, a conjecture which has been verified for the initial 1001 terms. Compare with A377096, 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+3) * Product_{k=1..n} (5*k + 2) for n >= 1 (conjecture).
EXAMPLE
G.f.: A(x) = 1 + x - x^2 - 7*x^3 + 74*x^4 - 371*x^5 + 579*x^6 + 8500*x^7 - 95812*x^8 + 505648*x^9 - 559784*x^10 - 16631826*x^11 + 179806704*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, -1, -7, 74, -371, 579, 8500, ...];
2: [1, 2, -1, -16, 135, -580, 317, 17864, ...];
3: [1, 3, 0, -26, 180, -645, -523, 26661, ...];
4: [1, 4, 2, -36, 207, -588, -1700, 33836, ...];
5: [1, 5, 5, -45, 215, -434, -3000, 38685, ...];
6: [1, 6, 9, -52, 204, -210, -4240, 40824, ...];
7: [1, 7, 14, -56, 175, 56, -5271, 40153, ...];
8: [1, 8, 20, -56, 130, 336, -5980, 36816, ...];
9: [1, 9, 27, -51, 72, 603, -6291, 31158, ...];
10: [1, 10, 35, -40, 5, 832, -6165, 23680, ...];
11: [1, 11, 44, -22, -66, 1001, -5599, 14993, ...];
12: [1, 12, 54, 4, -135, 1092, -4624, 5772, ...];
13: [1, 13, 65, 39, -195, 1092, -3302, -3289, ...];
14: [1, 14, 77, 84, -238, 994, -1722, -11520, ...];
15: [1, 15, 90, 140, -255, 798, 5, -18315, ...];
16: [1, 16, 104, 208, -236, 512, 1752, -23168, ...];
17: [1, 17, 119, 289, -170, 153, 3383, -25704, ...];
18: [1, 18, 135, 384, -45, -252, 4761, -25704, ...];
19: [1, 19, 152, 494, 152, -665, 5757, -23123, ...];
20: [1, 20, 170, 620, 435, -1036, 6260, -18100, 5, ...];
21: [1, 21, 189, 763, 819, -1302, 6188, -10959, -42315, 655564, ...];
22: [1, 22, 209, 924, 1320, -1386, 5500, -2200, -80696, 722568, ...];
23: [1, 23, 230, 1104, 1955, -1196, 4209, 7521, -111619, 722568, ...];
...
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, 1092, -25704, 722568, -23750496, 895732992, ...],
and appear to equal (-1)^n * (5*n+3) * Product_{k=1..n} (5*k + 2) 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) ) ); H=A; V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 04 2024
STATUS
approved