A372611 - OEIS

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A372611

Expansion of (1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x)).

1, 7, 26, 90, 310, 1082, 3844, 13892, 50950, 189130, 708876, 2677452, 10175356, 38863780, 149045960, 573559240, 2213551430, 8563950250, 33203854460, 128978378620, 501839077460, 1955475615820, 7629823818680, 29805375256120, 116558646378140, 456270710243332

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Conjecture: For p Pythagorean prime (A002144), a(p) - 7 == 0 (mod p).

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

LINKS

Table of n, a(n) for n=0..25.

FORMULA

a(n) = 6*A000984(n) - 5* A029759(n) = binomial(2*n,n) + 5*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).

a(n) = 2*a(n-1) + A028322(n) = 2*a(n-1) + binomial(2*n, n) + 3*binomial(2*n-2, n-1) for n >= 1.

a(n) = - 2^(n-1)*5*i + binomial(2*n,n)*(1-5/2*hypergeom([1, n + 1/2], [n + 1], 2)).

a(n) = 3*A082590(n-1) + A082590(n) for n >= 1.

a(n) = (7*A188622(n) - 4*A126966(n))/3.

a(n) = 2*A372239(n) - A372420(n).

MAPLE

a := n -> -2^(n-1)*5*I + binomial(2*n, n)*(1-5/2*hypergeom([1, n+1/2], [n+1], 2)): seq(simplify(a(n)), n = 0 .. 25);

PROG

(PARI) my(x='x+O('x^30)); Vec((1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ Michel Marcus, May 07 2024

CROSSREFS

Cf. A002144, A002145, A000984, A028322, A029759, A082590, A126966, A188622, A372239, A372420.

Sequence in context: A027138 A282703 A240256 * A279761 A245750 A268776

Adjacent sequences: A372608 A372609 A372610 * A372612 A372613 A372614

KEYWORD

nonn

AUTHOR

Mélika Tebni, May 07 2024

STATUS

approved