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Search: id:A107985
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| A107985 |
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Triangle read by rows: T(n,k)=(k+1)(n+2)(n-k+1)/2 for 0<=k<=n. |
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+0
3
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| 1, 3, 3, 6, 8, 6, 10, 15, 15, 10, 15, 24, 27, 24, 15, 21, 35, 42, 42, 35, 21, 28, 48, 60, 64, 60, 48, 28, 36, 63, 81, 90, 90, 81, 63, 36, 45, 80, 105, 120, 125, 120, 105, 80, 45, 55, 99, 132, 154, 165, 165, 154, 132, 99, 55, 66, 120, 162, 192, 210, 216, 210, 192, 162, 120
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Kekule numbers for certain benzenoids. Column 0 and the main diagonal yield the triangular numbers (A000217). Row sums yield A002415. T(n,n-k)=T(n,k), T(2n,n)=(n+1)^3.
T(n,k) = number of Dyck (n+3)-paths with 3 peaks (UDs) and last descent of length k+1. For example, T(1,1)=3 counts UUDUDUDD, UDUUDUDD, UDUDUUDD. The number of Dyck n-paths containing k peaks and with last descent of length j is j/n*binom[n,k-1]*binom[n-j-1,k-2] (where as usual binom[a,b]=0 for b<0 except that binom[ -1,-1]:=1). - David Callan (callan(AT)stat.wisc.edu), Jun 26 2006
As a rectangular array, A107985 is the accumulation array (cf. A144112) of the rectangular array W given by w(i,j)=i+j-1; i.e., W=A002024 as a rectangular array. [From Clark Kimberling (ck6(AT)evansville.edu), Sep 16 2008]
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 237, K{B(n,2,-l)}).
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EXAMPLE
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Triangle begins:
1;
3,3;
6,8,6;
10,15,15,10;
15,24,27,24,15;
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MAPLE
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T:=proc(n, k) if k<=n then (k+1)*(n+2)*(n-k+1)/2 else 0 fi end: for n from 0 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000217, A002415.
A002024. [From Clark Kimberling (ck6(AT)evansville.edu), Sep 16 2008]
Sequence in context: A050065 A078477 A098832 this_sequence A114999 A160733 A021752
Adjacent sequences: A107982 A107983 A107984 this_sequence A107986 A107987 A107988
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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 12 2005
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