A367725 Expansion of g.f. A(x) satisfying x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^5) such that A(0) = 1.

1, 1, 5, 30, 205, 1525, 12001, 98229, 827651, 7130614, 62528631, 556247554, 5007588460, 45535148222, 417625550140, 3858724742014, 35884576665516, 335616614245440, 3154800011439675, 29789198944740050, 282426795122071741, 2687467779597815314, 25658105671446219050

Offset

0,3

Links

Table of n, a(n) for n=0..22.

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)^5).

(2) x = (1+x)*A(x) - A(x)^2 + x*A(x)^5 such that A(0) = 1.

(3) A(x) = x / Series_Reversion(x*(1 + Series_Reversion( x/((1 + x)^5 + x) ))).

(4) a(n) = Sum_{k=1..n} binomial(n, k) * binomial(5*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 + 5*x^2 + 30*x^3 + 205*x^4 + 1525*x^5 + 12001*x^6 + 98229*x^7 + 827651*x^8 + 7130614*x^9 + 62528631*x^10 + ...
Let R(x) = x * (1 - x) / (1 - x - x^5) then R(A(x)) = x;
however, A(R(x)) does not equal x, rather
A(R(x)) = 1 + x + 5*x^2 + 30*x^3 + 205*x^4 + 1525*x^5 + 12002*x^6 + 98240*x^7 + 827752*x^8 + 7131535*x^9 + 62537177*x^10 + ...

Prog

(PARI) {a(n)=polcoeff(x/serreverse(x*(1+serreverse(x/((1 + x)^5 + 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(5*k-n, k-1))/(n+0^n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A243156, A367724.

Sequence in context: A091927 A253076 A165312 * A082301 A144180 A222050

Adjacent sequences: A367722 A367723 A367724 * A367726 A367727 A367728

Keyword

nonn

Author

Paul D. Hanna, Nov 28 2023

Status

approved