A059366 Triangle T(m,s), m >= 0, 0 <= s <= m, arising in the computation of certain integrals.
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
Offset: 0
Examples
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;
...
From _Petros Hadjicostas_, May 13 2020: (Start)
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)
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167; see a(m,s) (typo in a formula corrected below).
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
- Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see p. 85.
- 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.
- Konrad Jacobs, Discrete Stochastics, Springer Basel AG, 1992; see Section 2.1.
- Wikipedia, Arcsine distribution.
Crossrefs
Programs
-
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
-
Maple
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: seq(print(seq(A059366(m, s), s = 0..m)), m = 0..10) ; # Peter Bala, Apr 14 2024
-
Mathematica
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 *) -
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
Formula
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).]
From Reinhard Zumkeller, Apr 10 2004: (Start)
T(n,s) = T(n,n-s).
Sum_{s=0..n} T(n,s) = A000165(n). (End)
From Petros Hadjicostas, May 13 2020: (Start)
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)
From Peter Bala, Apr 14 2024: (Start)
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)
Extensions
More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001
Comments