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A028338
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Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x+2*n-1).
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14
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1, 1, 1, 3, 4, 1, 15, 23, 9, 1, 105, 176, 86, 16, 1, 945, 1689, 950, 230, 25, 1, 10395, 19524, 12139, 3480, 505, 36, 1, 135135, 264207, 177331, 57379, 10045, 973, 49, 1, 2027025, 4098240, 2924172, 1038016, 208054, 24640, 1708, 64, 1, 34459425
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OFFSET
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0,4
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COMMENTS
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Row sums are A000165. [Paul Barry, Feb 07 2009]
Exponential Riordan array (1/sqrt(1-2*x), log(1/sqrt(1-2*x))). [Paul Barry, May 09 2011]
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LINKS
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T. D. Noe, Rows n=0..50 of triangle, flattened
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FORMULA
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Triangle T(n, k), read by rows, given by [1, 2, 3, 4, 5, 6, 7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938 . - Philippe Deléham, Feb 20 2005
T(n, k) = sum(i=k..n, (-2)^(n-i) * binomial(i, k) * s(n, i)) where s(n, k) are signed Stirling numbers of the first kind. - Francis Woodhouse (fwoodhouse(AT)gmail.com), Nov 18 2005
G.f.: G.f.: 1/(1-(x+x*y)/(1-2*x/(1-(3*x+x*y)/(1-4*x/(1-(5*x+x*y)/(1-6*x*y/(1-... (continued fraction). [Paul Barry, Feb 07 2009]
a(n, m) = (2*n-1)*a(n-1,m) + a(n-1,m-1) with a(n, 0) = (2*n-1)!! and a(n, n) = 1. [Johannes W. Meijer, Jun 08 2009]
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EXAMPLE
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E.g. For n=4, (x + 1)*(x + 3)*(x + 5)*(x + 7) = x^4+16*x^3+86*x^2+176*x+105
1; 1,1; 3,4,1; 15,23,9,1; 105,176,86,16,1; ...
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MAPLE
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nmax:=8; for n from 0 to nmax do a(n, 0) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (2*n-1)*a(n-1, m) + a(n-1, m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # [Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012]
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MATHEMATICA
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T[n_, k_] := Sum[(-2)^(n-i) Binomial[i, k] StirlingS1[n, i], {i, k, n}] (Woodhouse)
Join[{1}, Flatten[Table[CoefficientList[Expand[Times@@Table[x+i, {i, 1, 2n+1, 2}]], x], {n, 0, 10}]]] (* Harvey P. Dale, Jan 29 2013 *)
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CROSSREFS
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A039757 is signed version.
Row sums: A000165.
Diagonals: A001147, A004041, A028339, A028340, A028341; A000012, A000290, A024196, A024197, A024198.
A161198 is a scaled triangle version and A109692 is a transposed triangle version.
Sequence in context: A059110 A100326 * A039757 A136228 A154829 A215241
Adjacent sequences: A028335 A028336 A028337 * A028339 A028340 A028341
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KEYWORD
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tabl,nonn,easy,nice
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AUTHOR
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R. W. Gosper
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STATUS
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approved
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