A367724 Expansion of g.f. A(x) satisfying x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^4) such that A(0) = 1.

1, 1, 4, 18, 90, 487, 2785, 16559, 101309, 633523, 4030964, 26012632, 169842909, 1119960451, 7447798584, 49891581704, 336355963821, 2280413834539, 15538054908152, 106345134539952, 730770552189188, 5039872364942277, 34872958276988909, 242026790074533733, 1684356151039022090

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

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

(2) x = (1+x)*A(x) - A(x)^2 + x*A(x)^4 such that A(0) = 1.

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

(4) a(n) = Sum_{k=1..n} binomial(n, k) * binomial(4*k-n, k-1))/n for n > 0 with a(0) = 1 (derived from a formula by Tani Akinari in A243156).

Example

G.f. A(x) = 1 + x + 4*x^2 + 18*x^3 + 90*x^4 + 487*x^5 + 2785*x^6 + 16559*x^7 + 101309*x^8 + 633523*x^9 + 4030964*x^10 + ...
Let R(x) = x * (1 - x) / (1 - x - x^4) then R(A(x)) = x;
however, A(R(x)) does not equal x, rather
A(R(x)) = 1 + x + 4*x^2 + 18*x^3 + 90*x^4 + 488*x^5 + 2794*x^6 + 16622*x^7 + 101732*x^8 + 636382*x^9 + 4050546*x^10 + ...

Prog

(PARI) {a(n)=polcoeff(x/serreverse(x*(1+serreverse(x/((1 + x)^4 + x +x*O(x^n))))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From a formula by Tani Akinari in A243156 */
{a(n)=0^n+sum(k=1, n, binomial(n, k)*binomial(4*k-n, k-1))/(n+0^n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A243156, A367725.

Sequence in context: A218760 A219305 A011270 * A355247 A269450 A206639

Adjacent sequences: A367721 A367722 A367723 * A367725 A367726 A367727

Keyword

nonn

Author

Paul D. Hanna, Nov 28 2023

Status

approved