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A096794
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Triangle read by rows: a(n,k) = number of Dyck n-paths such that number of DUs at level 1 plus number of UDs at level 2 is k, 0<=k<=n-1.
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0
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1, 0, 2, 1, 0, 4, 2, 4, 0, 8, 6, 8, 12, 0, 16, 18, 26, 24, 32, 0, 32, 57, 80, 84, 64, 80, 0, 64, 186, 260, 264, 240, 160, 192, 0, 128, 622, 864, 880, 768, 640, 384, 448, 0, 256, 2120, 2932, 2976, 2624, 2080, 1632, 896, 1024, 0, 512, 7338, 10112, 10248, 9024, 7280
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Column k has g.f. F(x)^(k+1)*(2y)^k where F(x)=(1-sqrt(1-4*x))/(3-sqrt(1-4*x)) is the g.f. for Fine's sequence A000957.
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FORMULA
| G.f. (1 - (1 - 4*x)^(1/2))/(3 - 2y + (2y-1)(1 - 4*x)^(1/2) ) = Sum_{n>=1, k>=0} a(n, k) x^n y^k.
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EXAMPLE
| Table begins
\ k 0, 1, 2, ...
n
1 | 1
2 | 0, 2
3 | 1, 0, 4
4 | 2, 4, 0, 8
5 | 6, 8, 12, 0, 16
6 | 18, 26, 24, 32, 0, 32
7 | 57, 80, 84, 64, 80, 0, 64
a(4,1) = 4 because UudUUDDD, UUUDDudD, UduUUDDD, UUUDDduD each contain one
relevant turn (in small type).
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CROSSREFS
| Row sums are the Catalan numbers A000108.
Sequence in context: A004558 A129699 A002349 * A106375 A194734 A201701
Adjacent sequences: A096791 A096792 A096793 * A096795 A096796 A096797
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KEYWORD
| nonn,tabl
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AUTHOR
| David Callan (callan(AT)stat.wisc.edu), Aug 17 2004
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