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A259095
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Triangle read by rows: T(n,r) = number of arrangements of n pennies in rows, with r contiguous pennies in the bottom row, and each higher row consisting of contiguous pennies, each touching two pennies in the row below (1 <= r <= n).
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7
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1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 1, 2, 4, 1, 0, 0, 0, 3, 3, 5, 1, 0, 0, 0, 2, 5, 4, 6, 1, 0, 0, 0, 1, 5, 7, 5, 7, 1, 0, 0, 0, 1, 5, 8, 9, 6, 8, 1, 0, 0, 0, 0, 4, 10, 11, 11, 7, 9, 1, 0, 0, 0, 0, 3, 11, 15, 14, 13, 8, 10, 1, 0, 0, 0, 0, 2, 9, 19, 20, 17, 15, 9, 11, 1, 0, 0, 0, 0, 1, 9, 20, 27, 25, 20, 17, 10, 12, 1, 0, 0, 0, 0, 1, 7, 20, 32, 35, 30, 23, 19, 11, 13, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,9
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COMMENTS
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LINKS
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FORMULA
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T(n,r) = Sum_{D(n,r)} Product_{k=2..m} abs(p[k]-p[k-1]) where the sum ranges over all partitions of n into distinct parts with maximal part r and the product over the m-1 pairs of successive parts; m is the number of parts in the partition under consideration. [Joerg Arndt, Apr 09 2016]
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EXAMPLE
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Triangle begins:
1,
0,1,
0,1,1,
0,0,2,1,
0,0,1,3,1,
0,0,1,2,4,1,
0,0,0,3,3,5,1,
0,0,0,2,5,4,6,1,
0,0,0,1,5,7,5,7,1,
0,0,0,1,5,8,9,6,8,1,
0,0,0,0,4,10,11,11,7,9,1,
0,0,0,0,3,11,15,14,13,8,10,1,
0,0,0,0,2,9,19,20,17,15,9,11,1,
0,0,0,0,1,9,20,27,25,20,17,10,12,1,
0,0,0,0,1,7,20,32,35,30,23,19,11,13,1,
...
(An unusually large number of rows are shown in order to explain the related sequences A005575-A005578.
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MAPLE
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b:= proc(n, i, d) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, 1, b(n, i-1, d+1)+`if`(i>n, 0, d*b(n-i, i-1, 1))))
end:
T:= (n, r)-> b(n-r, r-1, 1):
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MATHEMATICA
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b[n_, i_, d_] := b[n, i, d] = If[i*(i+1)/2 < n, 0, If[n == 0, 1, b[n, i-1, d+1] + If[i > n, 0, d*b[n-i, i-1, 1]]]];
T[n_, r_] := b[n-r, r-1, 1];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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