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A052712
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Expansion of e.g.f. (1 + 4*x - sqrt(1-8*x))/8.
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11
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0, 1, 2, 24, 480, 13440, 483840, 21288960, 1107025920, 66421555200, 4516665753600, 343266597273600, 28834394170982400, 2652764263730380800, 265276426373038080000, 28649854048288112640000
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OFFSET
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0,3
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COMMENTS
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Also the number of random walk labelings of the 2 X (n-1) king's graph, for n > 1. - Sela Fried, Apr 14 2023
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LINKS
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FORMULA
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D-finite with recurrence: a(0) = 0, a(1)=1, a(2)=2, a(n+1) = 4*(2*n-1)*a(n).
a(n) = 8^(n+1)*Gamma(n+3/2)/sqrt(Pi).
G.f.: (1 + 4*x - 2F0([1,-1/2], [], 8*x))/8. - R. J. Mathar, Jan 25 2020
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MAPLE
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spec := [S, {B=Prod(C, C), C=Union(B, S), S=Union(B, Z)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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Table[n!*2^(n-2)*CatalanNumber[n-1] +Boole[n==1]/2 +Boole[n==0]/4, {n, 0, 30}] (* G. C. Greubel, May 30 2022 *)
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PROG
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(SageMath) [2^(n-2)*factorial(n)*catalan_number(n-1) +bool(n==0)/8 +bool(n==1)/2 for n in (0..30)] # G. C. Greubel, May 30 2022
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CROSSREFS
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Cf. A052711, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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STATUS
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approved
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