A389246 - OEIS

%I #16 Oct 01 2025 04:12:06

%S 1,0,1,3,8,25,83,280,966,3402,12162,44011,160941,593814,2207816,

%T 8263724,31112838,117750432,447713309,1709407257,6551228310,

%U 25192992792,97181826623,375944600108,1458127244445,5669058115560,22089785423695,86251564121931,337422887622018

%N Expansion of (1/x) * Series_Reversion( x / (1 + x^2 / (1 - x)^3) ).

%H Vincenzo Librandi, <a href="/A389246/b389246.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(n+k-1,n-2*k).

%F a(n) = (1/(n+1)) * [x^n] (1 + x^2 / (1 - x)^3)^(n+1).

%F D-finite with recurrence -3*(3*n+2)*(3*n+1) *(5329638178135*n -6740091898537)*(n+1)*a(n) +(957297182797754*n^4 -2683976978115556*n^3 +2789459719303369*n^2 -911914488473519*n +40440551391222) *a(n-1) +(-2136238900546091*n^4 +15699572589884723*n^3 -39252401421363832*n^2 +39901025026117972*n -14020651307189502) *a(n-2) +6*(574225490711666*n^4 -4669351681023436*n^3 +14992495899086774*n^2 -21603111733355204*n +10995922036219955) *a(n-3) -(n-4) *(4457503378649747*n^3 -27859307256243307*n^2 +51102685535867403*n -25763046371721693) *a(n-4) +(n-4) *(n-5) *(4482834765302906*n^2 -23916955442793690*n +21414530310542049) *a(n-5) -31*(n-4) *(n-5) *(n-6) *(15703570527019*n -18045386290574) *a(n-6)=0. - _R. J. Mathar_, Sep 29 2025

%t Table[Coefficient[(1+x)^(n-1) (1+x^2+x^3)^(n+1),x,n]/(n+1),{n,0,25}] (* _Vincenzo Librandi_, Oct 01 2025 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x^2/(1-x)^3))/x)

%o (Magma) a := [ n eq 0 select 1 else (n eq 1 select 0 else &+[Binomial(n+1,k)*Binomial(n+k-1,3*k-1) : k in [1..Floor(n/2)]]/(n+1)) : n in [0..35] ];a; // _Vincenzo Librandi_, Oct 01 2025

%Y Cf. A161797, A389250.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Sep 26 2025