A352413 G.f. A(x) satisfies: A(x)^4 = (1-x) * (A(x) + x)^3.

1, 2, -6, 22, -115, 675, -4268, 28328, -194664, 1372984, -9882190, 72293094, -535946436, 4017628380, -30402186474, 231924537486, -1781685227700, 13771456884192, -107024198084814, 835752089556270, -6554617760700801, 51606615762239313

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

G.f. A(x) satisfies:

(1) A(x) = ( A(x)^4/(1-x) )^(1/3) - x.

(2) A(x)^4 = (1-x) * (A(x) + x)^3.

(3) A( x*(1+x)^3/(1 + x*(1+x)^3) ) = (1+x)^3/(1 + x*(1+x)^3).

(4) A(x) = x / Series_Reversion( x*(1+x)^3/(1 + x*(1+x)^3) ).

(5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 3, for n >= 1.

Example

G.f.: A(x) = 1 + 2*x - 6*x^2 + 22*x^3 - 115*x^4 + 675*x^5 - 4268*x^6 + 28328*x^7 - 194664*x^8 + ...
where A(x)^4 equals (1-x)*(A(x) + x)^3, as can be seen from the following power series expansions:
A(x)^4 = 1 + 8*x + 0*x^2 - 24*x^3 + 12*x^4 + 84*x^5 - 840*x^6 + 6816*x^7 - 52530*x^8 + ...
(A(x) + x)^3 = 1 + 9*x + 9*x^2 - 15*x^3 - 3*x^4 + 81*x^5 - 759*x^6 + 6057*x^7 - 46473*x^8 + ...
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,  2, -6,  22, -115, 675, -4268, 28328, ...];
n=2: [1,  4, -8,  20, -106, 626, -3972, 26424, ...];
n=3: [1,  6, -6,   2,  -45, 333, -2292, 15948, ...];
n=4: [1,  8,  0, -24,   12,  84,  -840,  6816, ...];
n=5: [1, 10, 10, -50,   25,   7,  -140,  1840, ...];
n=6: [1, 12, 24, -68,  -30, 102,   -44,   216, ...];
n=7: [1, 14, 42, -70, -161, 273,   -84,   -12, ...]; ...
in which the partial sum of row n up to column n equals (-1)^(n-1)*3, as illustrated by:
n=1:  3 = 1 +  2;
n=2: -3 = 1 +  4 + -8;
n=3:  3 = 1 +  6 + -6 +  2;
n=4: -3 = 1 +  8 +  0 + -24 +  12;
n=5:  3 = 1 + 10 + 10 + -50 +  25  +  7;
n=6: -3 = 1 + 12 + 24 + -68 + -30  + 102 +  -44;
n=7:  3 = 1 + 14 + 42 + -70 + -161 + 273 +  -84 + -12;
...

Prog

(PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^3/(1 + x*(1+x)^3 +x^2*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A352385, A352414, A352415.

Sequence in context: A111280 A095817 A370525 * A101042 A171339 A032266

Adjacent sequences: A352410 A352411 A352412 * A352414 A352415 A352416

Keyword

sign

Author

Paul D. Hanna, Mar 15 2022

Status

approved