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Search: id:A106831
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| A106831 |
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Define a triangle in which the entries are of the form +-1/(b!c!d!e!...), where the order of the factorials is important; read the triangle by rows and record and expand the denominators. Row n has 2^n terms. Row 0 is +1/2!. An entry +-1/b!c!d!... has two children, a left child -+1/(a+1)!b!c!... and a right child +-1/2!b!c!d!... |
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+0
1
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| 2, 6, 4, 24, 12, 12, 8, 120, 48, 36, 24, 48, 24, 24, 16, 720, 240, 144, 96, 144, 72, 72, 48, 240, 96, 72, 48, 96, 48, 48, 32, 5040, 1440, 720, 480, 576, 288, 288, 192, 720, 288, 216, 144, 288, 144, 144, 96, 1440, 480, 288, 192, 288, 144, 144, 96, 480, 192, 144, 96, 192
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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Let S_n = sum of entries in row n of the trriangle. Then for n>0, n!S_{n-1} is the Bernoulli number B_n.
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REFERENCES
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S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.
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EXAMPLE
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Triangle begins:
.........................+1..................
.........................--..................
.........................2!..................
..................-1...........+1............
..................--..........----...........
..................3!..........2!2!...........
..............+1.....-1....-1.........+1.....
..............--....---....----.....-----....
..............4!....2!3!...3!2!.....2!2!2!...
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MAPLE
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Contribution from Peter Luschny (peter(AT)luschny.de), Jun 12 2009: (Start)
The routine computes the triangle row by row and gives the numbers with
their sign. Thus A(1)=[2]; A(2)=[ -6, 4]; A(3)=[24, -12, -12, 8]; etc.
A := proc(n) local k, i, j, m, W, T; k := 2;
W := array(0..2^n); W[1] := [1, `if`(n=0, 1, 2)];
for i from 1 to n-1 do for m from k by 2 to 2*k-1 do
T := W[iquo(m, 2)]; W[m] := [ -T[1], T[2]+1, seq(T[j], j=3..nops(T))];
W[m+1] := [T[1], 2, seq(T[j], j=2..nops(T))]; od; k := 2*k; od;
seq(W[i][1]*mul(W[i][j]!, j=2..nops(W[i])), i=iquo(k, 2)..k-1) end:
seq(print(A(i)), i=1..5); (End)
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CROSSREFS
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Sequence in context: A069875 A019088 A096085 this_sequence A038212 A039656 A006233
Adjacent sequences: A106828 A106829 A106830 this_sequence A106832 A106833 A106834
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KEYWORD
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nonn,tabf,frac,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), May 22 2005
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EXTENSIONS
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More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Apr 28 2006
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