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A059366 Triangle T(m,s), m >= 0, 0 <= s <= m, arising in the computation of certain integrals.

%I A059366 #63 Apr 14 2024 13:47:53
%S A059366 1,1,1,3,2,3,15,9,9,15,105,60,54,60,105,945,525,450,450,525,945,10395,
%T A059366 5670,4725,4500,4725,5670,10395,135135,72765,59535,55125,55125,59535,
%U A059366 72765,135135,2027025,1081080,873180,793800,771750,793800,873180
%N A059366 Triangle T(m,s), m >= 0, 0 <= s <= m, arising in the computation of certain integrals.
%C A059366 From _Petros Hadjicostas_, May 12 2020: (Start)
%C A059366 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).
%C A059366 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)
%D A059366 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167; see a(m,s) (typo in a formula corrected below).
%H A059366 G. C. Greubel, <a href="/A059366/b059366.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A059366 Louis Comtet, <a href="https://www.jstor.org/stable/43667287">Fonctions génératrices et calcul de certaines intégrales</a>, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see p. 85.
%H A059366 Konrad Jacobs, <a href="https://doi.org/10.1007/978-3-642-95113-8_3">Das kombinatorische Äquivalenzprinzip und das arcsin-Gesetz von E. Sparre Andersen</a>, in: K. Jacobs (eds), Selecta Mathematica I, Heidelberger Taschenbücher, vol 49, Springer, Berlin, Heidelberg, 1969, pp. 53-81; see Lemma 3.3.
%H A059366 Konrad Jacobs, <a href="https://doi.org/10.1007/978-3-0348-8645-1">Discrete Stochastics</a>, Springer Basel AG, 1992; see Section 2.1.
%H A059366 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arcsine_distribution">Arcsine distribution</a>.
%F A059366 T(m+2, s) = (2*m+3)*(T(m+1, s-1) + T(m+1, s)) - 4*(m+1)^2*T(m, s-1).
%F A059366 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).]
%F A059366 From _Reinhard Zumkeller_, Apr 10 2004: (Start)
%F A059366 T(n,s) = A000984(s)*A000984(n-s)*A000142(n)/A000079(n).
%F A059366 T(n,s) = T(n,n-s).
%F A059366 Sum_{s=0..n} T(n,s) = A000165(n). (End)
%F A059366 From _Petros Hadjicostas_, May 13 2020: (Start)
%F A059366 T(m,s) = binomial(-1/2, s) * binomial(-1/2, m-s) * (-1)^m * m! * 2^m. [See Konrad (1992, pp. 189-190).]
%F A059366 T(m,m) = A001147(m) = T(m,0) for m >= 0.
%F A059366 T(m,m-1) = A001193(m-1) = T(m,1) for m >= 1.
%F A059366 T(m,m-2) = A001194(m) = T(m,2) for m >= 2.
%F A059366 T(m,m-3) = A001756(m) = T(m,3) for m >= 3.
%F A059366 T(m,floor(m/2)) = A001757(m) = T(m, ceiling(m/2)) for m >= 0.
%F A059366 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)
%F A059366 From _Peter Bala_, Apr 14 2024: (Start)
%F A059366 T(m, s) = (2*s - 1)*(m - s + 1)/(s*(2*m - 2*s + 1)) * T(m, s-1) for s >= 1.
%F A059366 T(m, s) = Sum_{i = 0..s} (-1)^(s-i)*binomial(m-i, s-i)*A368235(m, i). (End)
%e A059366 Triangle T(m,s) (with rows m >= 0 and columns 0 <= s <= m) begins as follows:
%e A059366     1;
%e A059366     1,   1;
%e A059366     3,   2,   3;
%e A059366    15,   9,   9,  15;
%e A059366   105,  60,  54,  60, 105;
%e A059366   945, 525, 450, 450, 525, 945;
%e A059366   ...
%e A059366 From _Petros Hadjicostas_, May 13 2020: (Start)
%e A059366 With m = 4, we have
%e A059366 J(4) = Integral_{t = 0..Pi/2} (A^2*cos^2(t) + B^2*sin^2(t))^(-4) dt
%e A059366 = Pi/(2^4*A*B*3!) * Sum_{s=0..3} T(3,s)*A^(-2*s)*B(-6+2*s)
%e A059366 = Pi/(96*A*B) * (15*B^(-6) + 9*A^(-2)*B^(-4) + 9*A^(-4)*B^(-2) + 15*A^(-6)). (End)
%p A059366 A059366 = proc(m, s) option remember; if s = 0  then (2*m)!/(2^m*m!) else
%p A059366 (2*s-1)*(m-s+1)/(s*(2*m-2*s+1)) * A059366(m, s-1) end if; end proc:
%p A059366 seq(print(seq(A059366(m, s), s = 0..m)), m = 0..10) ; # _Peter Bala_, Apr 14 2024
%t A059366 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 *)
%o A059366 (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
%o A059366 (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
%Y A059366 Central diagonal gives A001757. Other diagonals and columns include A001147, A001193, A001194.
%Y A059366 Cf. A000079, A000142, A000165, A000984, A368235.
%K A059366 tabl,nonn,easy
%O A059366 0,4
%A A059366 _N. J. A. Sloane_, Jan 28 2001
%E A059366 More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001