A394232 Expansion of 1 / sqrt(1 - 12*x + 28*x^2).

1, 6, 40, 288, 2184, 17136, 137536, 1121280, 9244768, 76866624, 643311360, 5412234240, 45729035520, 387757278720, 3297950730240, 28122838253568, 240356788028928, 2058330146595840, 17657749982408704, 151716952747917312, 1305392197618561024, 11245951765492015104, 96994770149873909760

Offset

0,2

Comments

3rd binomial transform of A001850.

Diagonal of the rational function 1 / (1 - x^2 - 6*x*y - 2*y^2).

Links

Table of n, a(n) for n=0..22.

Formula

E.g.f.: exp(6*x) * BesselI(0,2*sqrt(2)*x).

Coefficient of x^n in (1 + 6*x + 2*x^2)^n.

n * a(n) = 6 * (2*n-1) * a(n-1) - 28 * (n-1) * a(n-2) with a(0) = 1, a(1) = 6.

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * binomial(2*k,k) * 2^(n-k) * 3^(n-2*k).

Mathematica

nmax = 22; CoefficientList[Series[1/Sqrt[1 - 12 x + 28 x^2], {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Exp[6 x] BesselI[0, 2 Sqrt[2] x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[1] = 6; a[n_] := a[n] = (6 (2 n - 1) a[n - 1] - 28 (n - 1) a[n - 2])/n; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, 2 k] Binomial[2 k, k] 2^(n - k) 3^(n - 2 k), {k, 0, Floor[n/2]}], {n, 0, 22}]

Crossrefs

Cf. A001850, A006139, A059304, A080609, A084601, A098409, A098410, A154346, A394231, A394233.

Sequence in context: A390500 A005037 A081337 * A316912 A366199 A373622

Adjacent sequences: A394229 A394230 A394231 * A394233 A394234 A394235

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 13 2026

Status

approved