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Search: id:A033820
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| A033820 |
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Triangle read by rows: T(k,j) = ((2*j+1)/(k+1))*binomial(2*j,j)*binomial(2*k-2*j,k-j). |
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2
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| 1, 1, 3, 2, 4, 10, 5, 9, 15, 35, 14, 24, 36, 56, 126, 42, 70, 100, 140, 210, 462, 132, 216, 300, 400, 540, 792, 1716, 429, 693, 945, 1225, 1575, 2079, 3003, 6435, 1430, 2288, 3080, 3920, 4900, 6160, 8008, 11440, 24310, 4862, 7722, 10296, 12936, 15876, 19404
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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f(n,k)=2^{n-2(k-2)}sum(T(k-2,j)*binomial(n+2*(k-2-j),2*(k-2-j)),j=0..k-2) is the number of length n k-ary strings (k >= 2) which avoid a rising triple (pattern 123) or any other given 3-letter permutation pattern.
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LINKS
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Alexander Burstein, Enumeration of words with forbidden patterns, Ph.D. thesis, University of Pennsylvania, 1998.
Walter Shur, Two Game-Set Inequalities, J. Integer Seqs., Vol. 6, 2003.
Ira Gessel, Super ballot numbers.
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FORMULA
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T(k, 0)=binomial(2*k, k)/(k+1), the k-th Catalan number; T(k, k)=binomial(2*(k+1), k+1)/2, half the (k+1)-st central binomial coefficient sum of entries in row k (over j) = 2^{2*(k-1)}
T(k, j)=sum(C(k-i)D(i), i=0..j), C(i)=binomial(2*i, i)/(i+1), D(i)=binomial(2*i, i).
G.f.: 2/(1-4*x*y+sqrt((1-4*x)*(1-4*x*y))). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 14 2003
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EXAMPLE
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{1}, {1, 3}, {2, 4, 10}, {5, 9, 15, 35}, {14, 24, 36, 56, 126}, {42, 70, 100, 140, 210, 462}, {132, 216, 300, 400, 540, 792, 1716}, ...
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CROSSREFS
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Cf. A000108, A000984, A000302.
Essentially a reflected version of A078817.
Sequence in context: A083164 A094962 A084793 this_sequence A095259 A137824 A019321
Adjacent sequences: A033817 A033818 A033819 this_sequence A033821 A033822 A033823
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KEYWORD
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nonn,tabl
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AUTHOR
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Alexander Burstein (alexb(AT)math.upenn.edu)
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 10 2003
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