A084078 - OEIS

%I #23 Nov 16 2024 10:20:16

%S 1,2,4,10,24,66,172,498,1360,4066,11444,34970,100520,312066,911068,

%T 2862562,8457504,26824386,80006116,255680170,768464312,2471150402,

%U 7474561164,24161357010,73473471344,238552980386,728745517972

%N Length of list created by n substitutions k -> Range[-abs(k+1), abs(k-1), 2] starting with {0}.

%H G. C. Greubel, <a href="/A084078/b084078.txt">Table of n, a(n) for n = 0..1000</a>

%F a(2*n-1) = A027307(n), n >= 1.

%F a(n) = 2*A084075(n-1), n >= 1.

%F a(n) = ( 6*(35*n^2 -55*n -76)*a(n-1) + (275*n^4 -770*n^3 -203*n^2 +1736*n -912)*a(n-2) - 6*(5*n^2 +5*n -28)*a(n-3) + (n-4)*(n-2)*(25*n^2+5*n-48)*a(n-4) )/(n*(n+2)*(25*n^2 -45*n -28)), for n >= 4. - _G. C. Greubel_, Nov 24 2022

%F a(2*n) = A032349(n+1), n >= 0. - _Alexander Burstein_, Nov 19 2023

%e {0}, {-1,1}, {0,2,-2,0}, {-1,1,-3,-1,1,-1,1,3,-1,1}

%t Join[{1}, 2*Rest@CoefficientList[InverseSeries[Series[(-1 -6*n -8*n^2 + (1+ 2*n)^2*Sqrt[1+4*n])/(2*(n +4*n^2 +4*n^3)), {n, 0, 40}]], n]]

%t Length/@ Flatten/@ NestList[# /. k_Integer :> Range[-Abs[k+1], Abs[k-1], 2] &, {0}, 12]

%o (Magma) I:=[1,2,4,10]; [n le 4 select I[n] else (6*(35*n^2-125*n+14)*Self(n-1) + (275*n^4 -1870*n^3 +3757*n^2 -1268*n -1806)*Self(n-2) -6*(5*n^2-5*n-28)*Self(n-3) + (n-5)*(n-3)*(25*n^2-45*n-28)*Self(n-4))/((n-1)*(n+1)*(25*n^2-95*n+42)): n in [1..41]]; // _G. C. Greubel_, Nov 24 2022

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A084078

%o     if (n<4): return (1,2,4,10)[n]

%o     else: return (6*(35*n^2 -55*n -76)*a(n-1) +(275*n^4-770*n^3-203*n^2+1736*n-912)*a(n-2) -6*(5*n^2+5*n-28)*a(n-3) +(n-4)*(n-2)*(25*n^2+5*n-48)*a(n-4))/(n*(n+2)*(25*n^2-45*n-28))

%o [a(n) for n in range(41)] # _G. C. Greubel_, Nov 24 2022

%o (Python)

%o def replace(L): return [i for k in L for i in range(-abs(k + 1), 1 + abs(k - 1), 2)]

%o def aList(upto, L=[0]): return [1] + [len((L := replace(L))) for _ in range(upto)]

%o print(aList(12))  # _Peter Luschny_, Nov 16 2024

%Y Cf. A027307, A084075.

%K nonn

%O 0,2

%A _Wouter Meeussen_, May 11 2003