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A161736
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Denominators of the column sums of the BG2 matrix
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4
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1, 9, 75, 1225, 19845, 160083, 1288287, 41409225, 1329696225, 10667118605, 85530896451, 1371086188563, 21972535073125, 176021737014375, 1409850293610375, 90324408810638025, 5786075364399106425
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| 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, .. .
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.
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.
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.
The column sums sb(n) can be linked to other sequences, see the second Maple program.
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).
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FORMULA
| 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)).
a(n+1) = numerator of binomial( 2*n, n)^2 * n / 2^(n+1). - Michael Somos May 09 2011
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EXAMPLE
| sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..
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MAPLE
| restart; nmax:=10; 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)^k*A008956(n, n-k), k=0..n-m): od: sb(n+1):=sum(s(n, k), k=1..n)* 2/A001818(n); od: seq(sb(n), n=2..nmax);
restart; nmax:=10; for n from 0 to nmax do A001147(n):= doublefactorial(2*n-1) od: for n from 0 to nmax/2 do A133221(2*n+1):=A001147(n); A133221(2*n):=A001147(n) od: for n from 0 to nmax do A002474(n):=2^(2*n+1)*n!*(n+1)! od: for n from 1 to nmax do A161738(n):= ((product((2*n-3-2*k), k=0..floor(n/2-1)))) od: for n from 2 to nmax do sb(n):= A002474(n-2)/(A161738(n)*A133221(n-1))^2 od: seq(sb(n), n=2..nmax);
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PROG
| (PARI) {a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))} /* Michael Somos May 09 2011 */
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CROSSREFS
| Cf. Numerators A161737.
Cf. A001818, A008956, A013777, A134372 and A134375.
Cf. A001147, A133221, A161738 and A002474.
Sequence in context: A066222 A080254 A190916 * A056339 A056329 A190982
Adjacent sequences: A161733 A161734 A161735 * A161737 A161738 A161739
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KEYWORD
| easy,frac,nonn
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AUTHOR
| Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009
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