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A293653
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Young urn sequence (number of possible evolutions in n steps of the "Young" Pólya urn).
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2
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1, 2, 6, 30, 180, 1440, 12960, 142560, 1710720, 23950080, 359251200, 6107270400, 109930867200, 2198617344000, 46170964224000, 1061932177152000, 25486372251648000, 662645678542848000, 17891433320656896000, 518851566299049984000, 15565546988971499520000, 498097503647087984640000
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n+2) = (3/2)*a(n+1) + (9*n^2+21*n+12)/4*a(n), a(0) = 1, a(1) = 2.
Also, a(2n+k) has a hypergeometric expression (for k=0,1, see Maple code below) (proved).
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EXAMPLE
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We based the following explanations on Figure 1 from the Banderier et al. reference:
We have an urn with black and white balls in it.
At odd-numbered steps, we apply rule M1: we choose a ball, check its color, and add to the urn a ball of the same color.
At even-numbered steps, we apply rule M2: we choose a ball and check its color;
if it is black, we add 1 white ball and 1 black ball;
if it is white, we add 2 white balls.
At step 0, we start with the urn containing 1 black ball and 1 white ball (in short, 1b/1w). Accordingly, a(0)=1.
At step 1, we apply rule M1. This leads to 2 compositions: 2b/1w and 1b/2w. Accordingly, a(1) = 1 + 1 = 2.
At step 2, we apply rule M2. This leads to 3 compositions: 3b/2w, 2b/3w, and 1b/4w, each having multiplicity two. Accordingly, a(2) = 2 + 2 + 2 = 6. This leads to 4 compositions at step 3: 4b/2w (with multiplicity 6), and 3b/3w, 2b/4w, and 1b/5w (each of these last three with multiplicity 8). Accordingly, a(3) = 6 + 8 + 8 + 8 = 30.
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MAPLE
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a:=proc(n) if n mod 2= 0 then 3^n /GAMMA(2/3) * GAMMA(n/2+2/3) * GAMMA(n/2+1);
else 3^n /GAMMA(2/3) * GAMMA(n/2+7/6) * GAMMA(n/2+1/2); fi; end:
seq(a(n), n=0..30);
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MATHEMATICA
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f[n_] := FullSimplify[ 3^n/Gamma[2/3]*If[OddQ@ n, Gamma[n/2 + 7/6] Gamma[n/2 + 1/2], Gamma[n/2 + 2/3]Gamma[n/2 + 1]]]; Array[f, 20, 0] (* Robert G. Wilson v, Feb 07 2018 *)
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PROG
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(GAP) a:=[1, 2];; for n in [3..25] do a[n]:=(3/2)*a[n-1]+(9*(n-3)^2+21*n-51)/4*a[n-2]; od; a; # Muniru A Asiru, Dec 19 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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