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A084075 Length of list created by n substitutions k -> Range( -abs(k+1), abs(k-1), 2) starting with {1}. 5

%I #9 Nov 24 2022 05:22:39

%S 1,2,5,12,33,86,249,680,2033,5722,17485,50260,156033,455534,1431281,

%T 4228752,13412193,40003058,127840085,384232156,1235575201,3737280582,

%U 12080678505,36736735672,119276490193,364372758986,1187542872989

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

%H G. C. Greubel, <a href="/A084075/b084075.txt">Table of n, a(n) for n = 0..1000</a>

%F 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)).

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

%F 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

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

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

%t Length/@Flatten/@NestList[ #/.k_Integer:>Range[-Abs[k+1], Abs[k-1], 2] &, {1}, 8]

%o (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

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A084075

%o if (n<4): return (1,2,5,12)[n]

%o 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))

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

%Y Cf. A084076, A084077, A084078.

%Y Cf. A027307.

%K nonn

%O 0,2

%A _Wouter Meeussen_, May 11 2003

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