A352415 G.f. A(x) satisfies: A(x)^6 = (1-x) * (A(x) + x)^5.

1, 4, -15, 95, -815, 7881, -81946, 894100, -10097235, 117019845, -1383816039, 16631112009, -202545350505, 2494192904025, -31003553499195, 388498706407341, -4902323847971661, 62240419152427905, -794494411812382465, 10190561785036460125

Offset

0,2

Links

Table of n, a(n) for n=0..19.

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

Cf. A352385, A352413, A352414.

Sequence in context: A366697 A208991 A109365 * A079128 A356524 A289489

Adjacent sequences: A352412 A352413 A352414 * A352416 A352417 A352418

Keyword

sign

Author

Paul D. Hanna, Mar 15 2022

Status

approved