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A125232
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Triangle T(n,k) read by rows: the (n-k)-th term of the k-fold iterated partial sum of the pentagonal numbers.
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7
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1, 5, 1, 12, 6, 1, 22, 18, 7, 1, 35, 40, 25, 8, 1, 51, 75, 65, 33, 9, 1, 70, 126, 140, 98, 42, 10, 1, 92, 196, 266, 238, 140, 52, 11, 1, 117, 288, 462, 504, 378, 192, 63, 12, 1, 145, 405, 750, 966, 882, 570, 255, 75, 13, 1, 176, 550, 1155, 1716, 1848, 1452, 825, 330, 88, 14, 1
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table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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REFERENCES
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Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966, p 189.
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LINKS
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FORMULA
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G.f. as triangle: (1+2*x)/((1-x)^2*(1-x-x*y)). - Robert Israel, Nov 07 2016
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EXAMPLE
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First few rows of the triangle are:
1;
5, 1;
12, 6, 1;
22, 18, 7, 1;
35, 40, 25, 8, 1;
51, 75, 65, 33, 9, 1;
70, 126, 140, 98, 42, 10, 1;
...
Example: (5,3) = 65 = 25 + 40 = (4,3) + (4,2).
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MAPLE
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A125232 := proc(n, k) option remember ; if k = 0 then A000326(n) ; elif k = n-1 then 1 ; else procname(n-1, k)+procname(n-1, k-1) ; fi : end: # R. J. Mathar, Jun 09 2008
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MATHEMATICA
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nmax = 11; col[1] = Table[n(3n-1)/2, {n, 1, nmax}]; col[k_] := col[k] = Prepend[Accumulate[col[k-1]], 0]; Table[col[k][[n]], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2019 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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