A395675 Expansion of g^3/(2-g)^4, where g = 1+x*g^2 is the g.f. of A000108.

1, 7, 39, 198, 955, 4458, 20342, 91276, 404307, 1772610, 7707106, 33278292, 142853854, 610170148, 2594956620, 10994256152, 46425048451, 195456931506, 820725032042, 3438011713540, 14370725932458, 59951055922604, 249652366194484, 1037910552230568, 4308503499119150

Offset

0,2

Links

Table of n, a(n) for n=0..24.

Formula

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

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

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

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)).

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

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

From Seiichi Manyama, May 04 2026: (Start)

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

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

Prog

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

Crossrefs

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

Cf. A000108, A002457, A088218, A339240.

Sequence in context: A004057 A081898 A034267 * A198767 A026752 A026379

Adjacent sequences: A395672 A395673 A395674 * A395676 A395677 A395678

Keyword

nonn,easy

Author

Seiichi Manyama, May 03 2026

Status

approved