A361305 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^3.

1, 1, 5, 23, 123, 700, 4170, 25677, 162101, 1043603, 6825429, 45222437, 302892681, 2047499720, 13950769772, 95710823820, 660609751890, 4584018016679, 31960334260971, 223782306725768, 1572921720684820, 11094267854522250, 78499108540111380, 557041048588402170

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

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

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

(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * (1+x)^(3*n) / n!.

(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(3*n) / n! ).

(5) A(x) = x + Series_Reversion( Series_Reversion( x*(1+x)^(3/2) ) - x^2 )^2.

From Vaclav Kotesovec, Mar 09 2023: (Start)

Recurrence: 283*(n-3)*(n-2)*(n-1)*n*(3869140*n^3 - 39431172*n^2 + 133221959*n - 149076999)*a(n) = 4*(n-3)*(n-2)*(n-1)*(1199433400*n^4 - 14022813420*n^3 + 59620648652*n^2 - 107988096753*n + 68872774500)*a(n-1) + 6*(n-3)*(n-2)*(3718243540*n^5 - 52766330452*n^4 + 294066223701*n^3 - 803084308634*n^2 + 1072900001465*n - 559958090580)*a(n-2) + 12*(n-3)*(2453034760*n^6 - 43397123748*n^5 + 316599139024*n^4 - 1218191215329*n^3 + 2605017314614*n^2 - 2932345787601*n + 1355713586640)*a(n-3) + 5*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(3869140*n^3 - 27823752*n^2 + 65967035*n - 51417072)*a(n-4).

a(n) ~ 1/(2 * sqrt(Pi*(1 + s)*(1 + 8*s + 10*s^2)) * n^(3/2) * r^(n - 1/2)), where r = 0.1321273811013026086255933373480102325835852282463... and s = 0.2180852364825231879900920777342190033594997222087... are real roots of the system of equations r + s^2 * (1+s)^3 = s, s * (1+s)^2 * (2+5*s) = 1. (End)

a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(3*k,n-k)/(n+k+1). - Seiichi Manyama, Aug 24 2023

Example

G.f.: A(x) = x + x^2 + 5*x^3 + 23*x^4 + 123*x^5 + 700*x^6 + 4170*x^7 + 25677*x^8 + 162101*x^9 + 1043603*x^10 + ...
such that A(x) = x + A(x)^2 * (1 + A(x))^3.
Related series.
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 56*x^5 + 317*x^6 + 1876*x^7 + 11499*x^8 + 72352*x^9 + 464585*x^10 + ...
(1 + A(x))^3 = 1 + 3*x + 6*x^2 + 22*x^3 + 105*x^4 + 555*x^5 + 3151*x^6 + 18735*x^7 + 115200*x^8 + 726530*x^9 + ...

Prog

(PARI) {a(n)=polcoeff(serreverse(x-x^2*(1+x)^3+x*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(3*m)/m!)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*(1+x+x*O(x^n))^(3*m)/m!))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A361304, A361306, A214372.

Sequence in context: A359915 A362568 A121636 * A200028 A020032 A293088

Adjacent sequences: A361302 A361303 A361304 * A361306 A361307 A361308

Keyword

nonn

Author

Paul D. Hanna, Mar 08 2023

Status

approved