A395675 - OEIS

%I #22 May 04 2026 10:50:04

%S 1,7,39,198,955,4458,20342,91276,404307,1772610,7707106,33278292,

%T 142853854,610170148,2594956620,10994256152,46425048451,195456931506,

%U 820725032042,3438011713540,14370725932458,59951055922604,249652366194484,1037910552230568,4308503499119150

%N Expansion of g^3/(2-g)^4, where g = 1+x*g^2 is the g.f. of A000108.

%F G.f.: (Sum_{k>=0} binomial(2*k-1,k) * x^k) * (Sum_{k>=0} binomial(2*k,k) * x^k)^3.

%F G.f.: (1/2) * (1/(1-4*x)^(3/2) + 1/(1-4*x)^2).

%F Sum_{k>=1} a(k-1) * x^k/k^2 = (1/4) * log( Sum_{k>=0} binomial(2*k+2,k) * x^k ).

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

%F a(n) = A339240(n+1)/2.

%F E.g.f.: exp(2*x)*((1 + 4*x)*(exp(2*x) + BesselI(0, 2*x)) + 4*x*BesselI(1, 2*x))/2. - _Stefano Spezia_, May 03 2026

%F From _Seiichi Manyama_, May 04 2026: (Start)

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

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

%o (PARI) a(n) = (n+1)*(binomial(2*n+1,n)+2^(2*n))/2;

%Y Cf. A378504, A394853, A395664, A395665, A395667.

%Y Cf. A000108, A002457, A088218, A339240.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, May 03 2026