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A079308
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For a partition P of a positive integer, let f(P) be the product of k+1, over all parts k in P. Let a(n,r) be the sum of f(P) over all partitions P of n with smallest part r. Sequence gives table of a(n,r) for 1 <= r <= n, in the order a(1,1); a(2,1), a(2,2); a(3,1), a(3,2), a(3,3); ...
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2
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2, 4, 3, 14, 0, 4, 36, 9, 0, 5, 100, 12, 0, 0, 6, 236, 42, 16, 0, 0, 7, 602, 54, 20, 0, 0, 0, 8, 1368, 195, 24, 25, 0, 0, 0, 9, 3242, 246, 92, 30, 0, 0, 0, 0, 10, 7240, 759, 112, 35, 36, 0, 0, 0, 0, 11, 16386, 1134, 232, 40, 42, 0, 0, 0, 0, 0, 12, 35692, 2859, 528, 170, 48, 49, 0, 0, 0, 0, 0, 13
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OFFSET
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1,1
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LINKS
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EXAMPLE
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The partitions with minimal part 3 begin 3, 3+3, 4+3, 5+3, 6+3, 3+3+3, ... which yield the following values of f: 4, 16, 20, 24, 28, 64, ... therefore the 3rd column of our table begins 4,0,0,16,20,24,(28+64)=92,...
Triangle a(n,r) begins:
: 2;
: 4, 3;
: 14, 0, 4;
: 36, 9, 0, 5;
: 100, 12, 0, 0, 6;
: 236, 42, 16, 0, 0, 7;
: 602, 54, 20, 0, 0, 0, 8;
: 1368, 195, 24, 25, 0, 0, 0, 9;
: 3242, 246, 92, 30, 0, 0, 0, 0, 10;
: 7240, 759, 112, 35, 36, 0, 0, 0, 0, 11;
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MAPLE
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b:= proc(n, k) option remember; `if`(n=0, 1,
`if`(k>n, 0, b(n, k+1) +(k+1)*b(n-k, k)))
end:
a:= (n, k)-> b(n, k)-b(n, k+1):
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MATHEMATICA
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a[n_, r_] := Which[r>n, 0, r==n, n+1, True, a[n, r]=(r+1)Sum[a[n-r, s], {s, r, n-r}]]; Flatten[Table[a[n, r], {n, 1, 12}, {r, 1, n}]]
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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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