A352414 G.f. A(x) satisfies: A(x)^5 = (1-x) * (A(x) + x)^4.

1, 3, -10, 50, -345, 2681, -22416, 196700, -1786715, 16656155, -158443468, 1531830328, -15007700345, 148672680185, -1486712621330, 14987306377954, -152144993493979, 1554005064929735, -15958686622754240, 164676857033422880

Offset

0,2

Links

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

Formula

G.f. A(x) satisfies:

(1) A(x) = ( A(x)^5/(1-x) )^(1/4) - x.

(2) A(x)^5 = (1-x) * (A(x) + x)^4.

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

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

(5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 4, for n >= 1.

Example

G.f.: A(x) = 1 + 3*x - 10*x^2 + 50*x^3 - 345*x^4 + 2681*x^5 - 22416*x^6 + 196700*x^7 - 1786715*x^8 + ...
where A(x)^5 equals (1-x)*(A(x) + x)^4, as can be seen from the following power series expansions:
A(x)^5 = 1 + 15*x + 40*x^2 - 80*x^3 - 20*x^4 + 48*x^5 - 420*x^6 + 8160*x^7 - 109230*x^8 + ...
(A(x) + x)^4 = 1 + 16*x + 56*x^2 - 24*x^3 - 44*x^4 + 4*x^5 - 416*x^6 + 7744*x^7 - 101486*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,  3, -10,  50, -345, 2681, -22416, 196700, ...];
n=2: [1,  6, -11,  40, -290, 2292, -19346, 170784, ...];
n=3: [1,  9,  -3,  -3, -105, 1083, -10105,  94239, ...];
n=4: [1, 12,  14, -52,   21,  224,  -3208,  35792, ...];
n=5: [1, 15,  40, -80,  -20,   48,   -420,   8160, ...];
n=6: [1, 18,  75, -60, -255,  294,    -77,    720, ...];
n=7: [1, 21, 119,  35, -630,  350,    322,   -214, ...]; ...
in which the partial sum of row n up to column n equals (-1)^(n-1)*4, as illustrated by:
n=1:  4 = 1 +  3;
n=2: -4 = 1 +  6 + -11;
n=3:  4 = 1 +  9 +  -3 +  -3;
n=4: -4 = 1 + 12 +  14 + -52 +   21;
n=5:  4 = 1 + 15 +  40 + -80 +  -20 +  48;
n=6: -4 = 1 + 18 +  75 + -60 + -255 + 294 + -77;
n=7:  4 = 1 + 21 + 119 +  35 + -630 + 350 + 322 + -214;
...

Prog

(PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^4/(1 + x*(1+x)^4 +x^2*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A352385, A352413, A352415.

Sequence in context: A209902 A049370 A009343 * A341648 A307099 A048175

Adjacent sequences: A352411 A352412 A352413 * A352415 A352416 A352417

Keyword

sign

Author

Paul D. Hanna, Mar 15 2022

Status

approved