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Search: id:A121462
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| A121462 |
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Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having pyramid weight k (1<=k<=n). A pyramid in a Dyck word (path) is a factor of the form U^h D^h, where U=(1,1), D=(1,-1) and h is the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids. |
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4
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| 1, 0, 2, 0, 1, 4, 0, 1, 4, 8, 0, 1, 5, 12, 16, 0, 1, 6, 18, 32, 32, 0, 1, 7, 25, 56, 80, 64, 0, 1, 8, 33, 88, 160, 192, 128, 0, 1, 9, 42, 129, 280, 432, 448, 256, 0, 1, 10, 52, 180, 450, 832, 1120, 1024, 512, 0, 1, 11, 63, 242, 681, 1452, 2352, 2816, 2304, 1024, 0, 1, 12, 75, 316
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,n)=2^(n-1). Sum(k*T(n,k),k=1..n)=A030267(n)
Mirror image of triangle in A153342 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 31 2008]
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REFERENCES
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E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
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FORMULA
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T(n,k)=Sum(binomial(k-1,j)*binomial(n-k-1+j,j-1), j=0..k-1) for 2<=k<=n; T(1,1)=1; T(n,1)=0 for n>=2. G.f.=G=G(t,z)=tz(1-z)/(1-2tz-z+tz^2).
T(n+1,k+1)=A062110(n,k)*2^(2*k-n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 01 2006
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EXAMPLE
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T(4,3)=4 because we have (UD)U(UD)(UD)D, U(UD)(UD)(UD)D, U(UD)(UUDD)D and U(UUDD)(UD)D, where U=(1,1) and D=(1,-1) (the maximal pyramids are shown between parentheses).
Triangle starts:
1;
0,2;
0,1,4;
0,1,4,8;
0,1,5,12,16;
0,1,6,18,32,32;
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MAPLE
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T:=proc(n, k) if n=1 and k=1 then 1 elif k=1 then 0 elif k<=n then sum(binomial(k-1, j)*binomial(n-k-1+j, j-1), j=0..k-1) else 0 fi end: for n from 1 to 13 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001519, A030267, A091866.
Sequence in context: A099096 A099089 A121298 this_sequence A131487 A039991 A081265
Adjacent sequences: A121459 A121460 A121461 this_sequence A121463 A121464 A121465
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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), Jul 31 2006
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