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Denominators of the column sums of the BG2 matrix.
7

%I #45 Sep 08 2022 08:45:45

%S 1,9,75,1225,19845,160083,1288287,41409225,1329696225,10667118605,

%T 85530896451,1371086188563,21972535073125,176021737014375,

%U 1409850293610375,90324408810638025,5786075364399106425,46326420401234675625,370882277949065911875

%N Denominators of the column sums of the BG2 matrix.

%C 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 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 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 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 The column sums sb(n) can be linked to other sequences, see the second Maple program.

%C 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 a(n+2), for n >= 0, seems to coincide with the numerators belonging to A278145.- _Wolfdieter Lang_, Nov 16 2016

%H G. C. Greubel, <a href="/A161736/b161736.txt">Table of n, a(n) for n = 2..830</a>

%H R. Arratia, S. Garibaldi, 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 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 a(n+1) = numerator of C(2*n,n)^2 * n / 2^(n+1). - _Michael Somos_, May 09 2011

%F a(n) = A001902(2*n-3). - _Mats Granvik_, Nov 25 2018

%e sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

%p 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);

%p # End program 1

%p 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);

%p # End program 2

%p # Maple programs edited by _Johannes W. Meijer_, Sep 25 2012

%t 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 (PARI) {a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))}; /* _Michael Somos_, May 09 2011 */

%o (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 Cf. Numerators A161737.

%Y Cf. A001818, A001902, A008956, A013777, A134372 and A134375.

%Y Cf. A001147, A133221, A161738 and A002474.

%Y Cf. A278145.

%K easy,frac,nonn

%O 2,2

%A _Johannes W. Meijer_, Jun 18 2009