A063028 Reversion of x - x^2 + x^5.

0, 1, 1, 2, 5, 13, 35, 96, 264, 720, 1925, 4966, 12038, 25907, 41310, -5168, -468996, -2982240, -14350320, -61334790, -244951840, -934684465, -3447083370, -12365767620, -43304717625, -148314737961, -497033803314, -1628721662260, -5208556347700

Offset

0,4

Comments

For the reversion of x - a*x^2 - b*x^5 (a!=0, b!=0) we have a(n) = Sum_{j=0..floor((n-1)/3)} a^(n-4*j-1)*b^j*binomial(n-3*j-1, j)*binomial(2*n-3*j-2, n-1)/n, n > 0. - Vladimir Kruchinin, May 28 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..125

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Index entries for reversions of series

Formula

a(n) = Sum_{j=0..floor((n-1)/3)} (-1)^j*binomial(n-3*j-1, j)*binomial(2*n-3*j-2, n-1)/n, n > 0, a(0)=0. - Vladimir Kruchinin, May 28 2011

D-finite with recurrence +7576007*n*(n-1)*(n-2)*(n-3)*a(n) -14*(n-1)*(n-2)*(n-3)*(2435499*n+162464)*a(n-1) -70*(n-2)*(n-3)*(443090*n^2-7345575*n+16893064)*a(n-2) +1500*(n-3)*(126085*n^3-1478320*n^2+5789009*n-7573118)*a(n-3) +5*(8864375*n^4-23685000*n^3-505107125*n^2+2934387750*n-4417359408)*a(n-4) +250*(5*n-26)*(166625*n^3-1966175*n^2+7613615*n-9631377)*a(n-5) -131250*(5*n-27)*(5*n-31)*(5*n-24)*(5*n-28)*a(n-6)=0. - R. J. Mathar, Mar 21 2022

Mathematica

CoefficientList[InverseSeries[Series[y - y^2 + y^5, {y, 0, 30}], x], x]

Prog

(Maxima)
a(n):=sum((-1)^j*binomial(n-3*j-1, j)*binomial(2*n-3*j-2, n-1), j, 0, (n-1)/3)/n; /* Vladimir Kruchinin, May 28 2011 */
(PARI) x='x+O('x^66); /* that many terms */
Vec(serreverse(x-x^2+x^5)) /* show terms */ /* Joerg Arndt, May 28 2011 */

Crossrefs

Sequence in context: A000107 A366088 A370841 * A085810 A355040 A235611

Adjacent sequences: A063025 A063026 A063027 * A063029 A063030 A063031

Keyword

sign,easy

Author

Olivier Gérard, Jul 05 2001

Status

approved