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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Row sums are A000165. [From Paul Barry (pbarry(AT)wit.ie), Feb 07 2009]
Exponential Riordan array (1/sqrt(1-2*x), log(1/sqrt(1-2*x))). [Paul Barry, May 9 2011]
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LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
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FORMULA
| 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 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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). [From Paul Barry (pbarry(AT)wit.ie), Feb 07 2009]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)
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.
(End)
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EXAMPLE
| 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
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)
nmax:=8; mmax:=nmax: 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: T:=0; for n from 0 to nmax do for m from 0 to n do a(T):=a(n, m); T:=T+1 od: od: seq(a(n), n=0..T-1);
(End)
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MATHEMATICA
| T[n_, k_] := Sum[(-2)^(n-i) Binomial[i, k] StirlingS1[n, i], {i, k, n}] (Woodhouse)
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CROSSREFS
| 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. [Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009]
Sequence in context: A059110 A100326 * A039757 A136228 A154829 A055133
Adjacent sequences: A028335 A028336 A028337 * A028339 A028340 A028341
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KEYWORD
| tabl,nonn,easy,nice
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AUTHOR
| R. W. Gosper
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