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A329070
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Array read by ascending antidiagonals: T(n, k) = (k*n)!/(k^n*(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol.
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4
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1, 1, 1, 1, 2, 1, 1, 8, 6, 1, 1, 48, 180, 24, 1, 1, 384, 12960, 8064, 120, 1, 1, 3840, 1710720, 10644480, 604800, 720, 1, 1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1, 1, 645120, 109930867200, 244635697152000, 2303884477440000, 70355755008000, 10897286400, 40320, 1
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OFFSET
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0,5
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COMMENTS
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For information about the function W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)) (mentioned in the Formula section below), see Theorem 3.2 in Elizalde and Noy (2003) with u = 0 and m and a in the theorem equal to our m + 1. See also the documentation of array A327722.
By using the ratio test and the Stirling approximation to the gamma function, we may show that the radius of convergence of the power series for W_m(z) is infinity (for each m >= 0). Thus, the function W_m(z) (as defined by the above power series) is entire.
If we define S(m,s) = T(n-s, s+1) for m >= 0 and 0 <= s <= m, we get the triangular array that appears in the Example section below.
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LINKS
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FORMULA
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T(0,k) = 1, T(1,k) = k!, and T(2,k) = (2*k)!/(k + 1) for k >= 1.
T(n,1) = 1, T(n,2) = (2*n)!!, and T(n,3) is related to the Airy functions (see the documentation of A176730).
T(n+1,k) = (k-1)! * binomial(k*(n+1), k-1) * T(n,k) for n >= 0 and k >= 1.
T(n+1,k)/(k! * T(n,k)) = Cat(n+1, k), where Cat(d, k) = binomial(k*d, k)/(k * (d - 1) + 1) is a Fuss-Catalan number; see Theorem 1.2 in Schuetz and Whieldon (2014).
If F(k,z) = Sum_{n >= 0} z^(k*n)/T(n,k), then F(k,z) satisfies the o.d.e. F^(k-1)(k,z) - z*F(k,z) = 0.
If W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)), then 1/W_m(z) is the e.g.f. of row m of A327722(m,n), which counts permutations of [n] that avoid the consecutive pattern 12...(m+1)(m+3)(m+2) (or equivalently, the consecutive pattern (m+3)(m+2)...(3)(1)(2)).
The function W_m(z) satisfies the o.d.e. W_m^(m+2)(z) + z*W_m'(z) = 0 with W_m(0) = 1, W_m'(0) = -1, and W_m^(s)(0) = 0 for s = 2..(m + 1).
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EXAMPLE
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Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
1, 1, 1, 1, 1, 1, ...
1, 2, 6, 24, 120, 720, ...
1, 8, 180, 8064, 604800, 68428800, ...
1, 48, 12960, 10644480, 19813248000, 70355755008000, ...
...
Triangular array S(m,s) = T(m-s, s+1) (with rows m >= 0 and columns s >= 0):
1;
1, 1;
1, 2, 1;
1, 8, 6, 1;
1, 48, 180, 24, 1;
1, 384, 12960, 8064, 120, 1;
1, 3840, 1710720, 10644480, 604800, 720, 1;
1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1;
...
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MAPLE
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A := (n, k) -> `if`(k=0, 1, (GAMMA(1/k)*GAMMA(k*n+1))/(GAMMA(n+1/k)*k^n)):
seq(seq(A(n-k-1, k), k=1..n-1), n=0..10); # Peter Luschny, Nov 04 2019
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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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