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A182579
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Triangle read by rows: T(0,0) = 1, for n>0: T(n,n) = 2 and for k<=floor(n/2): T(n,2*k) = n/(n-k) * binomial(n-k,k), T(n,2*k+1) = (n-1)/(n-1-k) * binomial(n-1-k,k).
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8
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1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 2, 1, 1, 5, 4, 5, 2, 1, 1, 6, 5, 9, 5, 2, 1, 1, 7, 6, 14, 9, 7, 2, 1, 1, 8, 7, 20, 14, 16, 7, 2, 1, 1, 9, 8, 27, 20, 30, 16, 9, 2, 1, 1, 10, 9, 35, 27, 50, 30, 25, 9, 2, 1, 1, 11, 10, 44, 35, 77, 50, 55, 25, 11, 2
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OFFSET
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0,3
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COMMENTS
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A000204(n+1) = sum of n-th row, Lucas numbers;
A000204(n+3) = alternating row sum of n-th row;
A182584(n) = T(2*n,n), central terms;
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LINKS
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FORMULA
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T(n+1,2*k+1) = T(n,2*k), T(n+1,2*k) = T(n,2*k-1) + T(n,2*k).
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EXAMPLE
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Starting with 2nd row = [1 2] the rows of the triangle are defined recursively without computing explicitely binomial coefficients; demonstrated for row 8, (see also Haskell program):
. (0) 1 1 7 6 14 9 7 2 [A] row 7 prepended by 0
. 1 1 7 6 14 9 7 2 (0) [B] row 7, 0 appended
. 1 0 1 0 1 0 1 0 1 [C] 1 and 0 alternating
. 1 0 7 0 14 0 7 0 0 [D] = [B] multiplied by [C]
. 1 1 8 7 20 14 16 7 2 [E] = [D] added to [A] = row 8.
. 1 | 1
. 1 2 | 3
. 1 1 2 | 4
. 1 1 3 2 | 7
. 1 1 4 3 2 | 11
. 1 1 5 4 5 2 | 18
. 1 1 6 5 9 5 2 | 29
. 1 1 7 6 14 9 7 2 | 47
. 1 1 8 7 20 14 16 7 2 | 76
. 1 1 9 8 27 20 30 16 9 2 | 123
. 1 1 10 9 35 27 50 30 25 9 2 | 199 .
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MATHEMATICA
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T[_, 0] = 1;
T[n_, n_] /; n > 0 = 2;
T[_, 1] = 1;
T[n_, k_] := T[n, k] = Which[
OddQ[k], T[n - 1, k - 1],
EvenQ[k], T[n - 1, k - 1] + T[n - 1, k]];
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PROG
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(Haskell)
a182579 n k = a182579_tabl !! n !! k
a182579_row n = a182579_tabl !! n
a182579_tabl = [1] : iterate (\row ->
zipWith (+) ([0] ++ row) (zipWith (*) (row ++ [0]) a059841_list)) [1, 2]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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