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A369580
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a(n) := f(n, n), where f(0,0) = 1/3, f(0,k) = 0 and f(k,0) = 3^(k-1) if k > 0, and f(n, m) = f(n, m-1) + f(n-1, m) + 3*f(n-1, m-1) otherwise.
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1
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2, 16, 138, 1216, 10802, 96336, 861114, 7708416, 69072354, 619380496, 5557080938, 49879087296, 447852531986, 4022246329936, 36132550233498, 324645166734336, 2917340834679234, 26219438520320016, 235672871308226634, 2118552629658530496, 19046140604787232242, 171241206828437556816
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OFFSET
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1,1
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COMMENTS
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Take turns flipping a fair coin. The first to n heads wins. Sequence gives numerator of probability of first player winning. The denominator is .3^(2n-1).
It appears that a(n) for any n is divisible by 2^(A001511(n)).
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LINKS
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FORMULA
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Limit_{n->oo} a(n)/3^(2n-1) = 1/2.
a(n) = Sum_{i>=n} Sum_{j=0..n-1} binomial(i-1,n-1)*binomial(i-1,j)*3^(2n-1)/2^(2i-1).
9*a(n) - a(n+1) = 2*A162326(n) (conjectured).
a(n) = 3^(2n-1)*A(n, n) where A(0, k) = 0 for k > 0, A(k, 0) = 1 for k >= 0 and A(n, m) = (A(n-1, m) + A(n, m-1) + A(n-1, m-1))/3.
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PROG
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(Python)
def lis(n):
table = [[0]*(n+1) for _ in range(n+1)]
table[1][1] = 2
for i in range(1, n+1) :
table[i][0] = 3**(i-1)
for i in range(1, n+1) :
for j in range(1, n+1) :
if (i == 1 and j == 1) :
continue
table[i][j] = table[i][j-1] + table[i-1][j] + 3*table[i-1][j-1]
return [int(table[i][i]) for i in range(1, n+1)]
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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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