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A092905
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Triangle, read by rows, such that the partial sums of the n-th row form the n-th diagonal, for n>=0, where each row begins with 1.
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3
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1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 6, 4, 2, 1, 1, 6, 9, 7, 4, 2, 1, 1, 7, 12, 11, 7, 4, 2, 1, 1, 8, 16, 16, 12, 7, 4, 2, 1, 1, 9, 20, 23, 18, 12, 7, 4, 2, 1, 1, 10, 25, 31, 27, 19, 12, 7, 4, 2, 1, 1, 11, 30, 41, 38, 29, 19, 12, 7, 4, 2, 1, 1, 12, 36, 53, 53, 42, 30, 19, 12, 7, 4, 2, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums form A000070, which is the partial sums of the partition numbers (A000041). Rows read backwards converge to the row sums (A000070).
Contribution from Alford Arnold (Alford1940(AT)aol.com), Feb 07 2010: (Start)
The table can also be generated by summing sequences embedded within Table A008284
For example,
1 1 1 1 ... yields 1 2 3 4 ...
1 1 2 2 3 3 ... yields 1 2 4 6 9 12 ...
1 1 2 3 4 5 7 ... yields 1 2 4 7 11 16 ...
(End)
T(n,k) is also count of all 'replacable' cells in the (Ferrers plots of) the partitions on n in exactly k parts. [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 16 2010]
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FORMULA
| T(n, k) = sum_{j=0..k} T(n-k, j), with T(0, n) = 1 for all n>=0. A000070(n) = sum_{k=0..n} T(n, k).
E.g.f.: (1/(1-y))*(1/Product(1-x*y^k, k=1..infinity)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 29 2005
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EXAMPLE
| The third row is {1,3,2,1} and the third diagonal is the partial sums of the third row: {1,4,6,7,7,7,7,7,...}.
Rows begin:
{1},
{1,1},
{1,2,1},
{1,3,2,1},
{1,4,4,2,1},
{1,5,6,4,2,1},
{1,6,9,7,4,2,1},
{1,7,12,11,7,4,2,1},
{1,8,16,16,12,7,4,2,1},
{1,9,20,23,18,12,7,4,2,1},
{1,10,25,31,27,19,12,7,4,2,1},
{1,11,30,41,38,29,19,12,7,4,2,1},
{1,12,36,53,53,42,30,19,12,7,4,2,1},...
T(5,3)=4 because the partitions of 5 in exactly 3 parts are 221 and 311, and they give rise to partitions of 4 in four ways: 221->22 and 211, 311->211 and 31, since both their Ferrers plots have 2 'mobile cells' each. [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 16 2010]
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MAPLE
| T(n, k)=if(n<k|k<0, 0, if(n==k|k==0, 1, sum(j=0, min(k, n-k), T(n-k, j))))
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MATHEMATICA
| (*Needs["DiscreteMath`Combinatorica`"]; partitionexact[n_, m_] := TransposePartition /@ (Prepend[ #1, m] & ) /@ Partitions[n - m, m] *); mobile[p_?PartitionQ]:=1+Count[Drop[p, -1]-Rest[p], _?Positive]; Table[Tr[mobile/@partitionexact[n, k]], {n, 12}, {k, n}] [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 16 2010]
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CROSSREFS
| Antidiagonal sums form the partition numbers (A000041).
Cf. A000070.
Cf. A008284 [From Alford Arnold (Alford1940(AT)aol.com), Feb 07 2010]
Sequence in context: A077592 A194005 A055794 * A052509 A172119 A093628
Adjacent sequences: A092902 A092903 A092904 * A092906 A092907 A092908
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Mar 12 2004
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