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A028338
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Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
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27
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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, 71697105, 53809164, 20570444, 4574934, 626934, 53676, 2796, 81, 1
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OFFSET
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0,4
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COMMENTS
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Exponential Riordan array (1/sqrt(1-2*x), log(1/sqrt(1-2*x))). - Paul Barry, May 09 2011
The o.g.f.s D(d, x) of the column sequences, for d, d >= 0,(d=0 for the main diagonal) are P(d, x)/(1 - x)^(2*d+1), with the row polynomial P(d, x) = Sum_{m=0..d} A288875(d, m)*x^m. See A288875 for details. - Wolfdieter Lang, Jul 21 2017
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LINKS
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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. of row polynomials in y: 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
T(n, m) = (2*n-1)*T(n-1,m) + T(n-1,m-1) with T(n, 0) = (2*n-1)!! and T(n, n) = 1. - Johannes W. Meijer, Jun 08 2009
E.g.f. of row polynomials in y: (1/sqrt(1-2*x))*exp(-y*log(sqrt(1-2*x))) = exp(-(1+y)*log(sqrt(1-2*x))) = 1/sqrt(1-2*x)^(1+y).
E.g.f. of column m sequence: (1/sqrt(1-2*x))* (-log(sqrt(1-2*x)))^m/m!. For the special Sheffer, also known as exponential Riordan array, see a comment above. (End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 2^(n-1-p)*(1 + 2*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1). See a comment and references in A286718. - Wolfdieter Lang, Aug 09 2017
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EXAMPLE
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G.f. for n = 4: (x + 1)*(x + 3)*(x + 5)*(x + 7) = 105 + 176*x + 86*x^2 + 16*x^3 + x^4.
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9
0: 1
1: 1 1
2: 3 4 1
3: 15 23 9 1
4: 105 176 86 16 1
5: 945 1689 950 230 25 1
6: 10395 19524 12139 3480 505 36 1
7: 135135 264207 177331 57379 10045 973 49 1
8: 2027025 4098240 2924172 1038016 208054 24640 1708 64 1
9: 34459425 71697105 53809164 20570444 4574934 626934 53676 2796 81 1
...
row n = 10: 654729075 1396704420 1094071221 444647600 107494190 16486680 1646778 106800 4335 100 1.
O.g.f.s of diagonals d >= 0: D(2, x) = (3 + 8*x + x^2)/(1 - x)^5 generating [3, 23, 86, ...] = A024196(n+1), from the row d=2 entries of A288875 [3, 8, 1]. - Wolfdieter Lang, Jul 21 2017
Boas-Buck recurrence for column k=2 and n=4: T(4, 2) = (4!/2)*(2*(1+4*(5/12 )*T(2, 2)/2! + 1*(1 + 4*(1/2))*T(3,2)/3!) = (4!/2)*(8/3*1 + 3*9/3!) = 86. - Wolfdieter Lang, Aug 11 2017
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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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A161198 is a scaled triangle version and A109692 is a transposed triangle version.
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KEYWORD
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AUTHOR
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STATUS
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approved
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