A372239 Expansion of (1 + 2*x) / ((1 - 2*x)*sqrt(1 - 4*x)).

1, 6, 22, 76, 262, 916, 3260, 11800, 43334, 161028, 604052, 2283048, 8681116, 33171144, 127260088, 489870896, 1891057222, 7317881444, 28378110628, 110251755656, 429040567732, 1672032067544, 6524678847688, 25490986350416, 99696437839132, 390298689482216

Offset: 0

Author

Mélika Tebni, Apr 23 2024

Keywords

• nonn
• changed

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 6 == 0 (mod p).

Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 2 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028283, A029759, A082590, A126966, A188622.

Programs

Maple

a := n -> binomial(2*n,n) + 4*add(2^(n-k-1)*binomial(2*k,k), k = 0 .. n-1):
seq(a(n), n = 0 .. 25);
# Second program:
a:= proc(n) option remember; `if`(n=0,1,2*a(n-1)+2*binomial(2*n-2, n-1)*(3*n-1)/n) end: seq(a(n), n = 0 .. 25);
# Recurrence:
a := proc(n) option remember; if n < 2 then return [1, 6][n + 1] fi;
((-18*(n - 2)^2 - 42*n + 66)*a(n - 1) + 4*(3*n - 1)*(2*n - 3)*a(n - 2)) / (n*(4 - 3*n)) end: seq(a(n), n = 0..25);  # Peter Luschny, Apr 23 2024

Formula

a(n) = 5*A000984(n) - 4* A029759(n) = binomial(2*n,n) + 4*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).

a(n) = 2*a(n-1) + A028283(n) = 2*a(n-1) + 2*binomial(2n-2, n-1)*(3*n-1)/n for n >= 1.

a(n) = 2*A082590(n-1) + A082590(n) for n >= 1.

a(n) = 2*A188622(n) - A126966(n).

D-finite with recurrence n*a(n) +2*(-2*n-1)*a(n-1) +4*(-n+6)*a(n-2) +8*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 24 2024

E.g.f.: exp(2*x)*(BesselI(0, 2*x)*(1 + 4*x + 2*Pi*x*StruveL(1, 2*x)) - 2*Pi*x*BesselI(1, 2*x)*StruveL(0, 2*x)). - Stefano Spezia, Aug 29 2025