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A111910
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Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
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6
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1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 84, 30, 1, 1, 55, 330, 330, 55, 1, 1, 91, 1001, 2145, 1001, 91, 1, 1, 140, 2548, 10010, 10010, 2548, 140, 1, 1, 204, 5712, 37128, 68068, 37128, 5712, 204, 1, 1, 285, 11628, 116280, 352716, 352716, 116280, 11628, 285, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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Define a(n) = n!*(n+1/2)!*(n+1)!/(1/2)!.
S(n,k) = a(n+k)/(a(n)*a(k)) gives the sequence as a square array while T(n,k) = a(n)/(a(n-k)*a(k)) gives the sequence as a triangle.
S(n-1,k)*S(n,k+1)*S(n+1,k-1) = S(n-1,k+1)*S(n,k-1)*S(n+1,k). Cf. A091044.
(End)
As a number triangle:
T(n, k) = binomial(n+1, k)*binomial(2*n+1, 2*k)/((k+1)*(2*k+1)).
T(n, k) = binomial(2*n+1, 2*k)/((2*k+1)*binomial(n, k)) * A001263(n+1, k+1). (End)
As a triangle: T(n,k) = a(n)/(a(n-k)*a(k)), where a(n) = Product_{j = 1..n} s(p,j) with s(p,j) = Sum_{j = 1..n} j^p and p = 2. Note, p = 0 gives Pascal's triangle A007318, p = 1 gives the triangle of Narayana numbers A001263 and p = 3 gives the triangle A174158 whose entries are the squares of the Narayana numbers.
Let E(y) = Sum_{n >= 0} y^n/((n+1)!*(2*n+1)!). Then as a triangle this is the generalized Riordan array (E(y), y) as defined in Wang and Wang with respect to the sequence c_n = (n+1)!*(2*n+1)!. Cf. A001263.
Generating function: E(y)*E(x*y) = 1 + (1 + x)*y/(2!*3!) + (1 + 5*x + x^2)*y^2/(3!*5!) + (1 + 14*x + 14*x^2 + x^3)*y^3/(4!*7!) + ....
The n-th power of this array has a generating function E(y)^n*E(x*y). In particular, the matrix inverse has a generating function E(x*y)/E(y).
exp(y)*E(y) = 1 + 13*y/(2!*3!) + 421*y^2/(3!*5!) + 25368*y^3/(4!*7!) + ... is essentially a generating function for A081442. (End)
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EXAMPLE
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Array S(n,k) in rectangular form (n, k >= 0):
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 5, 14, 30, 55, 91, 140, 204, 285, ...
1, 14, 84, 330, 1001, 2548, 5712, 11628, 21945, ...
1, 30, 330, 2145, 10010, 37128, 116280, 319770, 793155, ...
1, 55, 1001, 10010, 68068, 352716, 1492260, 5393454, 17185025, ...
...
Array T(n,k) in triangular form (n >= 0 and 0 <= k <= n):
1,
1, 1,
1, 5, 1,
1, 14, 14, 1,
1, 30, 84, 30, 1,
1, 55, 330, 330, 55, 1,
1, 91, 1001, 2145, 1001, 91, 1,
...
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MAPLE
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a:=(p, q)->(p+q+1)!*(2*p+2*q+1)!/(p+1)!/(2*p+1)!/(q+1)!/(2*q+1)!: for n from 0 to 10 do seq(a(j, n-j), j=0..n) od; # yields sequence in triangular form
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MATHEMATICA
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Table[(# + q + 1)! (2 # + 2 q + 1)!/((# + 1)! (2 # + 1)! (q + 1)! (2 q + 1)!) &[r - q], {r, 0, 9}, {q, 0, r}] // Flatten (* Michael De Vlieger, Oct 21 2019 *)
Table[Binomial[n+1, k]*Binomial[2*n+1, 2*k]/((k+1)*(2*k+1)), {n, 0, 12}, {k, 0,
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PROG
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(Sage)
def A111910(n, k): return binomial(n+1, k)*binomial(2*n+1, 2*k)/((k+1)*(2*k+1))
(Magma)
T:= func< n, k | Binomial(n+1, k)*Binomial(2*n+1, 2*k)/((k+1)*(2*k+1)) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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