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Search: id:A108747
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| A108747 |
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Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n and having k returns to the x-axis. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). |
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+0
1
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| 2, 2, 4, 4, 8, 8, 10, 20, 24, 16, 28, 56, 72, 64, 32, 84, 168, 224, 224, 160, 64, 264, 528, 720, 768, 640, 384, 128, 858, 1716, 2376, 2640, 2400, 1728, 896, 256, 2860, 5720, 8008, 9152, 8800, 7040, 4480, 2048, 512, 9724, 19448, 27456, 32032, 32032, 27456
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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Row sums are the central binomial coefficients (A000984). T(n,1)=2C(n-1), where C(j)=binom(2j,j)/(j+1) is the j-th Catalan number (A000108). T(n,n)=2^n.
Triangle T(n,k), 1<=k<=n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 29 2005
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FORMULA
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T(n, k)=k2^k*binom(2n-k, n)/(2n-k) (1<=k<=n). G.f.=1/(1-2tzC), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
T(n, k) = 2^k*A106566(n, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 29 2005
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EXAMPLE
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T(2,2)=4 because we have u(d)u(d), u(d)d(u), d(u)d(u) and d(u)u(d) (return steps to x-axis shown between parentheses).
Triangle begins:
2;
2,4;
4,8,8;
10,20,24,16;
28,56,72,64,32;
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MAPLE
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T:=(n, k)->2^k*k*binomial(2*n-k, n)/(2*n-k): for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000984, A000108.
Sequence in context: A059867 A046971 A051754 this_sequence A116931 A145810 A034397
Adjacent sequences: A108744 A108745 A108746 this_sequence A108748 A108749 A108750
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 23 2005
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