OFFSET
0,2
FORMULA
G.f. A(x) satisfies:
(1) A(x) = ( A(x)^6/(1-x) )^(1/5) - x.
(2) A(x)^6 = (1-x) * (A(x) + x)^5.
(3) A( x*(1+x)^5/(1 + x*(1+x)^5) ) = (1+x)^5/(1 + x*(1+x)^5).
(4) A(x) = x / Series_Reversion( x*(1+x)^5/(1 + x*(1+x)^5) ).
(5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 5, for n >= 1.
EXAMPLE
G.f.: A(x) = 1 + 4*x - 15*x^2 + 95*x^3 - 815*x^4 + 7881*x^5 - 81946*x^6 + 894100*x^7 - 10097235*x^8 + ...
where A(x)^6 equals (1-x)*(A(x) + x)^5, as can be seen from the following power series expansions:
A(x)^6 = 1 + 24*x + 150*x^2 + 50*x^3 - 675*x^4 + 480*x^5 - 35*x^6 + 1980*x^7 + ...
(A(x) + x)^5 = 1 + 25*x + 175*x^2 + 225*x^3 - 450*x^4 + 30*x^5 - 5*x^6 + 1975*x^7 + ...
Related table.
Another defining property of the g.f. A(x) is illustrated here.
The table of coefficients of x^k in A(x)^n begins:
n=1: [1, 4, -15, 95, -815, 7881, -81946, 894100, ...];
n=2: [1, 8, -14, 70, -645, 6392, -67369, 741352, ...];
n=3: [1, 12, 3, -11, -210, 2793, -32964, 385869, ...];
n=4: [1, 16, 36, -84, 26, 504, -9506, 135524, ...];
n=5: [1, 20, 85, -85, -145, 129, -1050, 27550, ...];
n=6: [1, 24, 150, 50, -675, 480, -35, 1980, ...];
n=7: [1, 28, 231, 385, -1260, -399, 1708, -689, ...]; ...
in which the partial sum of row n up to column n equals (-1)^(n-1)*5, as illustrated by:
n=1: 5 = 1 + 4;
n=2: -5 = 1 + 8 + -14;
n=3: 5 = 1 + 12 + 3 + -11;
n=4: -5 = 1 + 16 + 36 + -84 + 26;
n=5: 5 = 1 + 20 + 85 + -85 + -145 + 129;
n=6: -5 = 1 + 24 + 150 + 50 + -675 + 480 + -35;
n=7: 5 = 1 + 28 + 231 + 385 + -1260 + -399 + 1708 + -689;
...
PROG
(PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^5/(1 + x*(1+x)^5 +x^2*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 15 2022
STATUS
approved