A063018 Reversion of x - x^2 - x^3 - x^4.

0, 1, 1, 3, 11, 44, 189, 850, 3951, 18832, 91542, 452075, 2261753, 11439372, 58394014, 300455892, 1556636807, 8113709916, 42518000652, 223868503324, 1183764310960, 6283573101960, 33470346433605, 178850415320010

Offset: 0

Author

Olivier Gérard, Jul 05 2001

Keywords

• nonn
• easy

Comments

For the reversion of x - a*x^2 - b*x^3 - c*x^4 (a!=0, b!=0, c!=0) we have a(n) = Sum(k=1,n-1, (Sum(j=0..k, a^(-n+3*k-j+1) * b^(n-3*k+2*j-1) * c^(k-j) * binomial(j,n-3*k+2*j-1) * binomial(k,j) ) ) * binomial(n+k-1,n-1))/n, n>1, a(1)=1. - Vladimir Kruchinin, May 28 2011

G.f. (with offset 1) satisfies A(x) = 1 + x*A(x)^2 + x^2*A(x)^3.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..105
• Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
• Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
• Index entries for reversions of series

Crossrefs

Cf. A001002 (reversion of y - y^2 - y^3).

Programs

Mathematica

CoefficientList[InverseSeries[Series[y - y^2 - y^3 - y^4, {y, 0, 30}], x], x]

Maxima

a(n):=sum((sum(binomial(j,n-3*k+2*j-1)*binomial(k,j),j,0,k))*binomial(n+k-1,n-1),k,1,n-1)/n; \\ Vladimir Kruchinin, May 28 2011

PARI

x='x+O('x^66); /* that many terms */
Vec(serreverse(x-x^2-x^3-x^4)) /* show terms */ /* Joerg Arndt, May 28 2011 */

Formula

a(n) = Sum(k=1..n-1, (Sum(j=0..k, binomial(j,n-3*k+2*j-1) * binomial(k,j))) * binomial(n+k-1,n-1))/n, n>1, a(1)=1, a(0)=0. - Vladimir Kruchinin, May 28 2011

D-finite with recurrence 2552*n*(n-1)*(n-2)*a(n) -4*(n-1)*(n-2)*(2909*n-3951)*a(n-1) -2*(n-2)*(6839*n^2 -31331*n +36576)*a(n-2) +(-17563*n^3 +138510*n^2 -359633*n +308670)*a(n-3) -120*(4*n-15)*(2*n-7)*(4*n-17)*a(n-4)=0. - R. J. Mathar, Mar 24 2023