A110616 revision #27
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A110616
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A convolution triangle of numbers based on A001764.
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5
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1, 1, 1, 3, 2, 1, 12, 7, 3, 1, 55, 30, 12, 4, 1, 273, 143, 55, 18, 5, 1, 1428, 728, 273, 88, 25, 6, 1, 7752, 3876, 1428, 455, 130, 33, 7, 1, 43263, 21318, 7752, 2448, 700, 182, 42, 8, 1, 246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1
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OFFSET
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0,4
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COMMENTS
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With offset 1 for n and k, T(n,k) = number of Dyck paths of semilength n for which all descents are of even length (counted by A001764) with no valley vertices at height 1 and with k returns to ground level. For example, T(3,2)=2 counts U^4 D^4 U^2 D^2, U^2 D^2 U^4 D^4 where U=upstep, D=downstep and exponents denote repetition. [David Callan, Aug 27 2009]
Riordan array (f(x), x*f(x)) with f(x) = (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))). - Philippe Deléham, Jan 27 2014
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LINKS
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FORMULA
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T(n, k) = Sum_{j, j>=0} T(n-1, k-1+j)*A000108(j); T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n.
GF: 1/(1 - x*y*TernaryGF) = 1 + (y)x + (y+y^2)x^2 + (3y+2y^2+y^3)x^3 +... where TernaryGF = 1 + x + 3x^2 + 12x^3 +... is the GF for A001764. [David Callan, Aug 27 2009]
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EXAMPLE
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Triangle begins:
1;
1, 1;
3, 2, 1;
12, 7, 3, 1;
55, 30, 12, 4, 1;
273, 143, 55, 18, 5, 1;
1428, 728, 273, 88, 25, 6, 1;
7752, 3876, 1428, 455, 130, 33, 7, 1;
43263, 21318, 7752, 2448, 700, 182, 42, 8, 1;
246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1;
...
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MATHEMATICA
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Table[(k + 1) Binomial[3 n - 2 k, 2 n - k]/(2 n - k + 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 28 2017 *)
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PROG
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(Maxima) T(n, k):=((k+1)*binomial(3*n-2*k, 2*n-k))/(2*n-k+1); // Vladimir Kruchinin, Nov 01 2011
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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proposed
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