A049128 Revert transform of x*(x - 1)^2/(1 - x + x^3).

1, 1, 2, 6, 20, 70, 255, 959, 3696, 14520, 57930, 234080, 955999, 3939949, 16364985, 68437033, 287910048, 1217627176, 5173854018, 22077273858, 94564541166, 406451008386, 1752472631360, 7577758539846, 32852955892191

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Index entries for reversions of series

Programs

Maple

a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 6][n],
      ((1404*n^4-9489*n^3+22155*n^2-21012*n+6840)*a(n-1)
       -(n-2)*(2548*n^3-14719*n^2+25575*n-12330)*a(n-2)
       +(n-2)*(n-3)*(2548*n^2-10663*n+7662)*a(n-3)
       -(23*(n-2))*(n-3)*(n-4)*(52*n-51)*a(n-4))/
       ((2*(2*n-1))*n*(52*n-103)*(n-3)))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 26 2015

Mathematica

a[n_] := 1/n*(Sum[Binomial[n, l]*Sum[Binomial[i-1, i-2*l]*Binomial[n-l, n-i-1], {i, 1, n-1}], {l, 1, n}])+1;
Array[a, 30] (* Jean-François Alcover, Apr 03 2017, after Vladimir Kruchinin *)

Maxima

a(n):=1/n*(sum(binomial(n,l)*sum(binomial(i-1,i-2*l)*binomial(n-l,n-i-1),i,1,n-1),l,1,n))+1; /* Vladimir Kruchinin, Jun 26 2015 */

Formula

a(n) = 1/n*(Sum_{l=1..n} C(n,l)*Sum_{i=1..n-1} C(i-1,i-2*l)*C(n-l,n-i-1)) + 1. - Vladimir Kruchinin, Jun 26 2015

D-finite with recurrence 2*n*(2*n-1) *a(n) +(-31*n^2+62*n-30) *a(n-1) +3*(20*n^2-54*n+25) *a(n-2) +(10*n^2-350*n+1023) *a(n-3) +(-124*n^2+1256*n-3063) *a(n-4) +(n-5) *(173*n-951)*a(n-5) -92*(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jul 20 2023

Extensions

NAME multiplied by x. - R. J. Mathar, Jul 23 2023