A084075 Length of list created by n substitutions k -> Range( -abs(k+1), abs(k-1), 2) starting with {1}.

1, 2, 5, 12, 33, 86, 249, 680, 2033, 5722, 17485, 50260, 156033, 455534, 1431281, 4228752, 13412193, 40003058, 127840085, 384232156, 1235575201, 3737280582, 12080678505, 36736735672, 119276490193, 364372758986, 1187542872989

Offset: 0

Author

Wouter Meeussen, May 11 2003

Keywords

• nonn

Examples

			{1}, {-2,0}, {-1,1,3,-1,1}, {0,2,-2,0,-4,-2,0,2,0,2,-2,0}
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A084076, A084077, A084078.

Cf. A027307, A215067, A034015 (even bisection).

Programs

Magma

I:=[1,2,5,12]; [n le 4 select I[n] else (6*(35*n^2-55*n-76)*Self(n-1) + (275*n^4-770*n^3-203*n^2+1736*n-912)*Self(n-2) -6*(5*n^2+5*n-28)*Self(n-3) + (n-4)*(n-2)*(25*n^2+5*n-48)*Self(n-4))/(n*(n+2)*(25*n^2-45*n-28)): n in [1..41]]; // G. C. Greubel, Nov 24 2022

Mathematica

Rest@CoefficientList[InverseSeries[Series[ (-1-6n-8n^2+(1+2n)^2 Sqrt[1+4n])/( 2(n+4n^2+4n^3)), {n, 0, 40}]], n]
Length/@Flatten/@NestList[ #/.k_Integer:>Range[-Abs[k+1], Abs[k-1], 2] &, {1}, 8]

Python

# replace iterates lists as described in Example.
def replace(L):
    return [i for k in L for i in range(-abs(k + 1), 1 + abs(k - 1), 2)]
def a(n):
  L = [1]
  for k in range(n): L=replace(L)
  return len(L)
print([a(n) for n in range(12)]) # F. Chapoton, Nov 15 2024

SageMath

@CachedFunction
def a(n): # a = A084075
    if n < 4: return (1, 2, 5, 12)[n]
    else: return (6*(35*n^2 +15*n -96)*a(n-1) +(275*n^4+330*n^3-863*n^2+120*n+126)*a(n-2) -6*(5*n^2+15*n-18)*a(n-3) +(n-3)*(n-1)*(25*n^2+55*n-18)*a(n-4))/((n+1)*(n+3)*(25*n^2+5*n-48))
[a(n) for n in range(41)] # G. C. Greubel, Nov 24 2022

Formula

G.f. is the series reversion of (-1 -6*x -8*x^2 + (1+2*x)^2 * sqrt(1+4*x))/(2*(x +4*x^2 +4*x^3)).

a(2*n) = A027307(n)/2, n >= 1.

a(n) = ( 6*(35*n^2 +15*n -96)*a(n-1) + (275*n^4 +330*n^3 -863*n^2 +120*n +126)*a(n-2) - 6*(5*n^2 +15*n -18)*a(n-3) + (n-3)*(n-1)*(25*n^2 +55*n -18)*a(n-4) )/((n+1)*(n+3)*(25*n^2 +5*n -48)), n >= 4. - G. C. Greubel, Nov 24 2022