A393656 - OEIS

%I #16 Mar 25 2026 08:59:29

%S 1,2,5,14,44,156,611,2550,11001,48224,213325,950820,4271352,19346480,

%T 88335740,406334918,1881301815,8759367346,40982288984,192561037552,

%U 908216648372,4298359134020,20407008376965,97164626491380,463863430765254,2219912272685016,10647915018993791

%N Expansion of (1/x) * Series_Reversion( x * ( Sum_{k=0..3} (-x)^k )^2 ).

%H Vincenzo Librandi, <a href="/A393656/b393656.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1/x) * Series_Reversion( x * ((1-x^4) / (1+x))^2 ).

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n+k+1,k) * binomial(3*n-2*k+1,n-2*k).

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(2*n+k+1,k) * binomial(2*n+2,n-4*k).

%t CoefficientList[Normal@Series[InverseSeries@Series[x*((1-x^4)/(1+x))^2,{x,0,50}]/x,{x,0,27}],x] (* _Vincenzo Librandi_, Mar 24 2026 *)

%o (PARI) a(n) = sum(k=0, n\4, binomial(2*n+k+1, k)*binomial(2*n+2, n-4*k))/(n+1);

%o (Magma) N := 25; R<x> := PowerSeriesRing(Rationals(), N+5); f:= x*((1-x^4)/(1+x))^2; g:= Reverse(f) div x; Coeffs := [Coefficient(g,i):i in [0..N]]; Coeffs; // _Vincenzo Librandi_, Mar 24 2026

%Y Cf. A063019, A393657.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 24 2026