A299501 Expansion of (1 - 6*x + 7*x^2 - 2*x^3 + x^4)^(-1/2).

1, 3, 10, 37, 145, 588, 2437, 10251, 43582, 186785, 805585, 3492064, 15200753, 66399763, 290910490, 1277803957, 5625184321, 24811849020, 109631120869, 485153695995, 2149941422590, 9539307910561, 42374000475457, 188421560848512, 838633172823745, 3735857124917763

Offset: 0

Author

Peter Luschny, Feb 15 2018

Keywords

• nonn

Comments

See A299500 for a family of related polynomials.

Crossrefs

Cf. A249946, A299500.

Programs

Maple

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

Mathematica

CoefficientList[Series[(1 - 6 x + 7 x^2 - 2 x^3 + x^4 )^(-1/2), {x, 0, 25}], x]

Formula

a(n) = Sum_{k=0..n} 2^(n-k)*binomial(n, k)*hypergeom([-k, k-n, k-n], [1, -n], 1/2).

D-finite with recurrence: (-2+n)*a(-4+n) + (-2*n+3)*a(n-3) + (7*n-7)*a(-2+n) + (-6*n+3)*a(-1+n) + n*a(n) = 0.

A249946(n) = a(n) - 2*a(n-1) + a(n-2) for n >= 2.