A372420 Expansion of (1 + x) / ((1 - 2*x)*sqrt(1 - 4*x)).

1, 5, 18, 62, 214, 750, 2676, 9708, 35718, 132926, 499228, 1888644, 7186876, 27478508, 105474216, 406182552, 1568563014, 6071812638, 23552366796, 91525132692, 356242058004, 1388588519268, 5419533876696, 21176597444712, 82834229300124, 324326668721100

Offset: 0

Author

Mélika Tebni, Apr 30 2024

Keywords

• nonn

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 5 == 0 (mod p).

Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028270, A029759, A082590, A126966, A188622.

Programs

Maple

a := n -> -2^(n-1)*3*I + binomial(2*n, n)*(1-3/2*hypergeom([1, n+1/2], [n+1], 2)):
seq(simplify(a(n)), n = 0 .. 25);

PARI

my(x='x+O('x^40)); Vec((1 + x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ Michel Marcus, Apr 30 2024

Formula

a(n) = 4*A000984(n) - 3* A029759(n) = binomial(2*n,n) + 3*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).

a(n) = 2*a(n-1) + A028270(n) = 2*a(n-1) + binomial(2*n, n) + binomial(2*n-2, n-1) for n >= 1.

a(n) = - 2^(n-1)*3*i + binomial(2*n,n)*(1-3/2*hypergeom([1,n+1/2],[n + 1],2)).

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

a(n) = (5*A188622(n) - 2*A126966(n)) / 3.

D-finite with recurrence n*a(n) -5*n*a(n-1) +2*(n+5)*a(n-2) +4*(2*n-5)*a(n-3)=0. - R. J. Mathar, May 01 2024