A301476 Expansion of (sqrt(8*x^2 - 4*x + 1)*(1 - 4*x))^(-1).

1, 6, 26, 100, 374, 1428, 5636, 22728, 92134, 371844, 1491148, 5954936, 23755068, 94842824, 379167304, 1517205392, 6072429190, 24299685732, 97210253564, 388810290840, 1555033861204, 6219560062424, 24877574062264, 99512092354800, 398060511603676, 1592275849685736

Offset

0,2

Links

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

Formula

a(n) = -(sqrt(2)*(-4)^n/n!)*d^(n+1)[arcsinh(1/x), 1] where d^n[f(x), x0] denotes the n-th derivative of f at x = x0.

D-finite with recurrence: a(n) = ((24*n + 56)*a(n+1) - (8*n + 22)*a(n+2) + (n + 3)*a(n+3)) / (32*n + 64).

Maple

A301476_list := proc(len) local ogf, ser;
ogf := (sqrt(8*x^2 - 4*x +1)*(1-4*x))^(-1); ser := series(ogf, x, len+1):
seq(coeff(ser, x, n), n=0..len-1) end: A301476_list(26);
# Alternatively:
a := n -> subs(x=1, -sqrt(2)*(-4)^n/n!*diff(arcsinh(1/x), x$(n+1))):
seq(a(n), n=0..25);

Mathematica

RecurrenceTable[{a[n] == ((24 n + 56) a[n + 1] - (8 n + 22) a[n + 2] + (n + 3) a[n + 3]) / (32 n + 64), a[0] == 1, a[1] == 6, a[2] == 26}, a, {n, 0, 25}]

Crossrefs

Sequence in context: A094811 A005022 A125107 * A290347 A034560 A307306

Adjacent sequences: A301473 A301474 A301475 * A301477 A301478 A301479

Keyword

nonn,easy

Author

Peter Luschny, Apr 06 2018

Status

approved