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Search: id:A035324
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| A035324 |
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A convolution triangle of numbers, generalizing Pascal's triangle A007318. |
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+0
16
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| 1, 3, 1, 10, 6, 1, 35, 29, 9, 1, 126, 130, 57, 12, 1, 462, 562, 312, 94, 15, 1, 1716, 2380, 1578, 608, 140, 18, 1, 6435, 9949, 7599, 3525, 1045, 195, 21, 1, 24310, 41226, 35401, 19044, 6835, 1650, 259, 24, 1, 92378, 169766, 161052, 97954, 40963, 12021, 2450
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Replacing each '2' in the recurrence by '1' produces Pascal's triangle A007318(n-1,m-1). The columns appear as A001700, A008549, A045720, A045894,...
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LINKS
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W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.
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FORMULA
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a(n+1, m) = 2*(2*n+m)*a(n, m)/(n+1) + m*a(n, m-1)/(n+1), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1; G.f. for column m: ((x*c(x)/sqrt(1-4*x))^m)/x, where c(x) = g.f. for Catalan numbers A000108. a(n, m)=: s2(3; n, m).
With offset 0( 0<=k<=n), T(n,k)=Sum_{j, j>=0}A039598(n,j)*binomial(j,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007
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EXAMPLE
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{1}; {3,1}; {10,6,1}; {35,29,9,1};...
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CROSSREFS
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Cf. A000108, A007318. Row sums: A049027(n), n >= 1.
If offset 0 (n >= m >= 0): convolution triangle based on A001700 (central binomial coeffs. of odd order).
Alternating row sums give A000108 (Catalan numbers).
Sequence in context: A057967 A132964 A134283 this_sequence A091965 A107056 A116384
Adjacent sequences: A035321 A035322 A035323 this_sequence A035325 A035326 A035327
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KEYWORD
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easy,nice,nonn,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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