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A059366
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Triangle T(m,s), m >= 0, 0 <= s <= m, arising in the computation of certain integrals.
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6
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1, 1, 1, 3, 2, 3, 15, 9, 9, 15, 105, 60, 54, 60, 105, 945, 525, 450, 450, 525, 945, 10395, 5670, 4725, 4500, 4725, 5670, 10395, 135135, 72765, 59535, 55125, 55125, 59535, 72765, 135135, 2027025, 1081080, 873180, 793800, 771750, 793800, 873180
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OFFSET
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0,4
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COMMENTS
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Following Comtet (1974, pp. 166-167), let J(m) = Integral_{t = 0..Pi/2} (A^2*cos^2(t) + B^2*sin^2(t))^(-m)) dt for m >= 0. Then J(m+1) = (Pi/(2^(m+1)*A*B*m!)) * Sum_{s=0..m} T(m,s)*A^(-2*s)*B^(-2*m+2*s).
Given m >= 0, the collection of numbers T(m,s)/A000165(m) = T(m,s)/(m!*2^m), s = 0..m, forms a discrete probability distribution on the set {0,1,...,m}, which is known as the "finite discrete arcsine distribution of order m". See Konrad (1969, Section 3.3) and Konrad (1992, Section 2.1, pp. 189-190). (End)
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167; see a(m,s) (typo in a formula corrected below).
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LINKS
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Konrad Jacobs, Das kombinatorische Äquivalenzprinzip und das arcsin-Gesetz von E. Sparre Andersen, in: K. Jacobs (eds), Selecta Mathematica I, Heidelberger Taschenbücher, vol 49, Springer, Berlin, Heidelberg, 1969, pp. 53-81; see Lemma 3.3.
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FORMULA
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T(m+2, s) = (2*m+3)*(T(m+1, s-1) + T(m+1, s)) - 4*(m+1)^2*T(m, s-1).
T(m, s) = m!*Sum_{k=0..s} (-1)^k*2^(2*k-m)*binomial(s, k)*binomial(2*m-2*k, s)*binomial(2*m-2*k-s, m-k). [Typo in Comtet (1974, p. 166) corrected by Petros Hadjicostas, May 12 2020, using Comtet (1967, p. 85).]
T(n,s) = T(n,n-s).
Sum_{s=0..n} T(n,s) = A000165(n). (End)
T(m,s) = binomial(-1/2, s) * binomial(-1/2, m-s) * (-1)^m * m! * 2^m. [See Konrad (1992, pp. 189-190).]
T(m,m) = A001147(m) = T(m,0) for m >= 0.
T(m,m-1) = A001193(m-1) = T(m,1) for m >= 1.
T(m,m-2) = A001194(m) = T(m,2) for m >= 2.
T(m,m-3) = A001756(m) = T(m,3) for m >= 3.
T(m,floor(m/2)) = A001757(m) = T(m, ceiling(m/2)) for m >= 0.
Lim_{m -> infinity} Sum_{s: s/m <= x} T(m,s)/A000165(m) = (2/Pi)*arcsin(sqrt(x)) for x in [0,1], where the summation is over those s in {0,1,...,m} that satisfy s/m <= x. (End)
T(m, s) = (2*s - 1)*(m - s + 1)/(s*(2*m - 2*s + 1)) * T(m, s-1) for s >= 1.
T(m, s) = Sum_{i = 0..s} (-1)^(s-i)*binomial(m-i, s-i)*A368235(m, i). (End)
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EXAMPLE
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Triangle T(m,s) (with rows m >= 0 and columns 0 <= s <= m) begins as follows:
1;
1, 1;
3, 2, 3;
15, 9, 9, 15;
105, 60, 54, 60, 105;
945, 525, 450, 450, 525, 945;
...
With m = 4, we have
J(4) = Integral_{t = 0..Pi/2} (A^2*cos^2(t) + B^2*sin^2(t))^(-4) dt
= Pi/(2^4*A*B*3!) * Sum_{s=0..3} T(3,s)*A^(-2*s)*B(-6+2*s)
= Pi/(96*A*B) * (15*B^(-6) + 9*A^(-2)*B^(-4) + 9*A^(-4)*B^(-2) + 15*A^(-6)). (End)
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MAPLE
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A059366 = proc(m, s) option remember; if s = 0 then (2*m)!/(2^m*m!) else
(2*s-1)*(m-s+1)/(s*(2*m-2*s+1)) * A059366(m, s-1) end if; end proc:
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MATHEMATICA
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Table[Binomial[2*s, s]*Binomial[2*n - 2*s, n - s]*n!/2^n, {n, 0, 10}, {s, 0, n}] // Flatten (* G. C. Greubel, Jan 08 2017 *)
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PROG
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(PARI) for(n=0, 10, for(s=0, n, print1(binomial(2*s, s)*binomial(2*n - 2*s, n - s)*n!/2^n, ", "))) \\ G. C. Greubel, Jan 08 2017
(Magma) /* as triangle */ [[Binomial(2*s, s)*Binomial(2*n-2*s, n-s)*Factorial(n)/2^n: s in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 09 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001
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STATUS
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approved
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