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A216973
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Exponential Riordan array [x*exp(x),x].
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4
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0, 1, 0, 2, 2, 0, 3, 6, 3, 0, 4, 12, 12, 4, 0, 5, 20, 30, 20, 5, 0, 6, 30, 60, 60, 30, 6, 0, 7, 42, 105, 140, 105, 42, 7, 0, 8, 56, 168, 280, 280, 168, 56, 8, 0, 9, 72, 252, 504, 630, 504, 252, 72, 9, 0, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 0
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OFFSET
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0,4
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COMMENTS
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This is the triangle of denominators from Leibniz's harmonic triangle, A003506, augmented with a main diagonal of 0's.
Note, the usual definition of the exponential Riordan array [f(x), x*g(x)] associated with a pair of power series f(x) and g(x) requires f(0) to be nonzero. Here we don't make this assumption. - Peter Bala, Feb 13 2017
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LINKS
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FORMULA
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T(n,k) = (n-k)*binomial(n,k) for 0 <= k <= n.
E.g.f.: x*exp(x)*exp(x*t) = 1 + x + (2 + 2*t)*x^2/2! + (3 + 6*t + 3*t^2)*x^3/3! + ....
The exponential Riordan array [x*exp(x),x] factors as [x,x]*[exp(x),x] = A132440*A007318.
This array is the infinitesimal generator for A116071; that is, Exp(A216973) = A116071, where Exp denotes the matrix exponential. A signed version of the array is the infinitesimal generator for A215652.
The first column of the array Exp(t*A216973) is the sequence of idempotent polynomials, the row polynomials of A059297.
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EXAMPLE
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Triangle begins
.n\k.|..0.....1.....2.....3.....4.....5.....6
= = = = = = = = = = = = = = = = = = = = = = =
..0..|..0
..1..|..1.....0
..2..|..2.....2.....0
..3..|..3.....6.....3.....0
..4..|..4....12....12.....4.....0
..5..|..5....20....30....20.....5.....0
..6..|..6....30....60....60....30.....6.....0
...
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MAPLE
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A216973_row := proc(n) x*exp(x)*exp(x*t): series(%, x, n+1): n!*coeff(%, x, n):
seq(coeff(%, t, k), k=0..n) end:
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MATHEMATICA
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(* The function RiordanArray is defined in A256893. *)
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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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