%I #24 May 25 2024 16:13:16
%S 1,1,2,6,20,70,255,959,3696,14520,57930,234080,955999,3939949,
%T 16364985,68437033,287910048,1217627176,5173854018,22077273858,
%U 94564541166,406451008386,1752472631360,7577758539846,32852955892191
%N Revert transform of x*(x - 1)^2/(1 - x + x^3).
%H Alois P. Heinz, <a href="/A049128/b049128.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F 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
%F 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
%p a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 6][n],
%p ((1404*n^4-9489*n^3+22155*n^2-21012*n+6840)*a(n-1)
%p -(n-2)*(2548*n^3-14719*n^2+25575*n-12330)*a(n-2)
%p +(n-2)*(n-3)*(2548*n^2-10663*n+7662)*a(n-3)
%p -(23*(n-2))*(n-3)*(n-4)*(52*n-51)*a(n-4))/
%p ((2*(2*n-1))*n*(52*n-103)*(n-3)))
%p end:
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Jun 26 2015
%t 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;
%t Array[a, 30] (* _Jean-François Alcover_, Apr 03 2017, after _Vladimir Kruchinin_ *)
%o (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 */
%K nonn
%O 1,3
%A _Olivier Gérard_
%E NAME multiplied by x. - _R. J. Mathar_, Jul 23 2023