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A161736 Denominators of the column sums of the BG2 matrix.

%I A161736 #59 Feb 13 2025 08:27:28
%S A161736 1,9,75,1225,19845,160083,1288287,41409225,1329696225,10667118605,
%T A161736 85530896451,1371086188563,21972535073125,176021737014375,
%U A161736 1409850293610375,90324408810638025,5786075364399106425,46326420401234675625,370882277949065911875,5938020471163465810125
%N A161736 Denominators of the column sums of the BG2 matrix.
%C A161736 The BG2 matrix coefficients, see also A008956, are defined by BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .
%C A161736 A different way to define the matrix coefficients is BG2[2*m,n] = (1/m)*sum(LAMBDA(2*m-2*k,n-1)*BG2[2*k,n], k=0..m-1) with LAMBDA(2*m,n-1) = (1-2^(-2*m))*zeta(2*m)-sum((2*k-1)^(-2*m), k=1..n-1) and BG2[0,n] = Pi/2 for m = 0, 1, 2, .. , and n = 1, 2, 3 .. , with zeta(m) the Riemann zeta function.
%C A161736 The columns sums of the BG2 matrix are defined by sb(n) = sum(BG2[2*m,n], m=0..infinity) for n = 2, 3, .. . For large values of n the value of sb(n) approaches Pi/2.
%C A161736 It is remarkable that if we assume that BG2[2m,1] = 2 for m = 0, 1, .. the columns sums of the modified matrix converge to the original sb(n) values. The first Maple program makes use of this phenomenon and links the sb(n) with the central factorial numbers A008956.
%C A161736 The column sums sb(n) can be linked to other sequences, see the second Maple program.
%C A161736 We observe that the column sums sb(n) of the BG2(n) matrix are related to the column sums sl(n) of the LG2(n) matrix, see A008956, by sb(n) = (-1)^(n+1)*(2*n-1)*sl(n).
%C A161736 a(n+2), for n >= 0, seems to coincide with the numerators belonging to A278145. - _Wolfdieter Lang_, Nov 16 2016
%C A161736 Suppose that, given values f(x-2*n+1), f(x-2*n+3), ..., f(x-1), f(x+1), ..., f(x+2*n-3), f(x+2*n-1), we approximate f(x) using the first 2*n terms of its Taylor series. Then 1/sb(n+1) is the coefficient of f(x-1) and f(x+1). - _Matthew House_, Dec 03 2024
%H A161736 G. C. Greubel, <a href="/A161736/b161736.txt">Table of n, a(n) for n = 2..830</a>
%H A161736 R. Arratia, S. Garibaldi, and J. Kilian, <a href="http://arxiv.org/abs/1310.7055">Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process</a>,  arXiv:1310.7055 [math.PR], 2013.
%F A161736 a(n) = denom(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161737(n) = numer(sb(n)).
%F A161736 a(n+1) = numerator of C(2*n,n)^2 * n / 2^(n+1). - _Michael Somos_, May 09 2011
%F A161736 a(n) = A001902(2*n-3). - _Mats Granvik_, Nov 25 2018
%F A161736 a(n) = numerator((1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - _Peter Luschny_, Feb 13 2025
%e A161736 sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..
%p A161736 nmax := 18; for n from 0 to nmax do A001818(n) := (doublefactorial(2*n-1))^2 od: for n from 0 to nmax do A008956(n, 0):=1 od: for n from 0 to nmax do A008956(n, n) := A001818(n) od: for n from 1 to nmax do for m from 1 to n-1 do A008956(n, m) := (2*n-1)^2*A008956(n-1, m-1) + A008956(n-1, m) od: od: for n from 1 to nmax do for m from 0 to n do s(n, m):=0; s(n, m) := s(n, m)+ sum((-1)^k1*A008956(n, n-k1), k1=0..n-m): od: sb1(n+1) := sum(s(n, k1), k1=1..n) * 2/A001818(n); od: seq(sb1(n), n=2..nmax); # End program 1
%p A161736 nmax1 := nmax; for n from 0 to nmax1 do A001147(n):= doublefactorial(2*n-1) od: for n from 0 to nmax1/2 do A133221(2*n+1) := A001147(n); A133221(2*n) := A001147(n) od: for n from 0 to nmax1 do A002474(n) := 2^(2*n+1)*n!*(n+1)! od: for n from 1 to nmax1 do A161738(n) := ((product((2*n-3-2*k1), k1=0..floor(n/2-1)))) od: for n from 2 to nmax1 do sb2(n) := A002474(n-2) / (A161738(n)*A133221(n-1))^2 od: seq(sb2(n), n=2..nmax1); # End program 2
%p A161736 # Above Maple programs edited by _Johannes W. Meijer_, Sep 25 2012
%p A161736 r := n -> (1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> numer(simplify(r(n))):
%p A161736 seq(a(n), n = 1..21);  # _Peter Luschny_, Feb 12 2025
%t A161736 sb[2]=2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2; Table[sb[n] // Denominator, {n, 2, 20}] (* _Jean-François Alcover_, Aug 14 2017 *)
%o A161736 (PARI) {a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))}; /* _Michael Somos_, May 09 2011 */
%o A161736 (Magma) [Denominator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // _G. C. Greubel_, Sep 26 2018
%Y A161736 Cf. Numerators A161737.
%Y A161736 Cf. A001818, A001902, A008956, A013777, A134372 and A134375.
%Y A161736 Cf. A001147, A133221, A161738 and A002474, A278145.
%K A161736 easy,frac,nonn
%O A161736 2,2
%A A161736 _Johannes W. Meijer_, Jun 18 2009