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Search: id:A129696
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| A129696 |
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Antidiagonal sums of triangular array T defined in A014430: T(j,k) = Binomial(j+1, k)-1 for 1 <= k <= j. |
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+0
3
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| 1, 2, 5, 9, 17, 29, 50, 83, 138, 226, 370, 602, 979, 1588, 2575, 4171, 6755, 10935, 17700, 28645, 46356, 75012, 121380, 196404, 317797, 514214, 832025, 1346253, 2178293, 3524561, 5702870, 9227447, 14930334, 24157798, 39088150, 63245966
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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First differences 1, 3, 4, 8, 12, ... are A052952(n), n >= 1.
Row sums of T are in A125128.
If T is contrued as a lower triangular matrix M over the rational field, the inverse M^-1 is a lower triangular matrix containing fractions. Its row sums are the Bernoulli numbers. First column of M^-1 is 1, -1, 2/3, -1/4, -1/30, 1/12, 1/42, -1/12, ... . Multiplied by j! this gives 1, -2, 4, -6, -4, 60, 120, -3660, ... .
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REFERENCES
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P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969.
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FORMULA
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a(n)=sum{k=0..floor(n/2), A000071(n-2k+3)}=sum{k=0..floor(n/2), sum{j=0..n-2k, F(j+1)}}. [From Paul Barry (pbarry(AT)wit.ie), Jan 18 2009]
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EXAMPLE
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First seven rows of T are
[ 1 ]
[ 2, 2 ]
[ 3, 5, 3 ]
[ 4, 9, 9, 4 ]
[ 5, 14, 19, 14, 5 ]
[ 6, 20, 34, 34, 20, 6 ]
[ 7, 27, 55, 69, 55, 27, 7 ].
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PROGRAM
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(MAGMA) m:=36; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=1 to j do M[j, k]:=Binomial(j+1, k)-1; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; /* Klaus Brockhaus, Jun 11 2007 */
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CROSSREFS
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Cf. A014430, A052952, A125128, A027641.
Sequence in context: A081996 A034329 A133470 this_sequence A082281 A000569 A115851
Adjacent sequences: A129693 A129694 A129695 this_sequence A129697 A129698 A129699
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KEYWORD
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nonn
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AUTHOR
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Paul Curtz (bpcrtz(AT)free.fr), Jun 01 2007
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EXTENSIONS
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Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 11 2007
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