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A193229
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A double factorial triangle.
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3
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1, 1, 1, 3, 3, 2, 15, 15, 12, 6, 105, 105, 90, 60, 24, 945, 945, 840, 630, 360, 120, 10395, 10395, 9450, 7560, 5040, 2520, 720, 135135, 135135, 124740, 103950, 75600, 45360, 20160, 5040, 2027025, 2027025, 1891890, 1621620, 1247400, 831600, 453600, 181440, 40320
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OFFSET
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0,4
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COMMENTS
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The double factorial triangle coefficients are T(n,k), n >= 0 and 0 <= k <= n.
The T(n,0) equal the double factorial numbers A001147(n) = (2*n-1)!!.
The T(n,n) equal the factorial numbers A000142(n) = n!.
The row sums equal the double factorial numbers A000165(n) = (2*n)!!.
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LINKS
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FORMULA
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T(n,k) = the (k+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = i, i >= 1 and 1 <= j <= i+1, and M[i,j] = 0, i >= 1 and j >= i+2, see the examples.
It appears that T(n,k) = (2*n-k)!/(2^(n-k)*(n-k)!) with conjectural e.g.f. 1/(x*(1-2*z) + (1-x)*sqrt(1-2*z)) = 1 + (1+x)*z + (3+3*x+2*x^2)*z^2/2! + .... Cf. A102625. - Peter Bala, Jul 09 2012
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EXAMPLE
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The first few rows of matrix M[i,j] are:
1, 1, 0, 0, 0, 0, ...
2, 2, 2, 0, 0, 0, ...
3, 3, 3, 3, 0, 0, ...
4, 4, 4, 4, 4, 0, ...
5, 5, 5, 5, 5, 5, ...
The first few rows of triangle T(n,k) are:
1;
1, 1;
3, 3, 2;
15, 15, 12, 6;
105, 105, 90, 60, 24;
945, 945, 840, 630, 360, 120;
10395, 10395, 9450, 7560, 5040, 2520, 720;
135135, 135135, 124740, 103950, 75600, 45360, 20160, 5040;
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MAPLE
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nmax:=7: M := Matrix(1..nmax+1, 1..nmax+1): for i from 1 to nmax do for j from 1 to i+1 do M[i, j] := i od: od: for n from 0 to nmax do B := M^n: for k from 0 to n do T(n, k) := B[1, k+1] od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jul 21 2011
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PROG
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(PARI) row(n)=(matrix(n, n, i, j, (i>j-2)*i)^(n-1))[1, ] \\ M. F. Hasler, Jul 24 2011
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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