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A035324
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A convolution triangle of numbers, generalizing Pascal's triangle A007318.
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20
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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; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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, A035330...
Triangle T(n,k), 1<=k<=n, given by (0, 3/1, 1/3, 5/3, 3/5, 7/5, 5/7, 9/7, 7/9, 11/9, 9/11, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe, Jan 28 2012
Riordan array (1, c(x)/sqrt(1-4x)) where c(x) = g.f. for Catalan numbers A000108, first column (k = 0) omitted . - DELEHAM Philippe, Jan 28 2012
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LINKS
| 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
| 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
T(n+1,n) = 3*n = A008585(n).
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EXAMPLE
| {1}; {3,1}; {10,6,1}; {35,29,9,1};...
Triangle (0,3,1/3,5/3,3/5,...) DELTA (1,0,0,0,0,0, ...) begins :
1
0, 1
0, 3, 1
0, 10, 6, 1
0, 35, 29, 9, 1
0, 126, 130, 57, 12, 1
0, 462, 562, 312, 94, 15, 1
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CROSSREFS
| 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: A171509 A171505 A134283 * A171814 A091965 A171568
Adjacent sequences: A035321 A035322 A035323 * A035325 A035326 A035327
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KEYWORD
| easy,nice,nonn,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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