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A098568
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Triangle of triangular binomial coefficients, read by rows, where column k has the g.f.: 1/(1-x)^((k+1)*(k+2)/2) for k>=0.
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11
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1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 56, 55, 15, 1, 1, 21, 126, 220, 120, 21, 1, 1, 28, 252, 715, 680, 231, 28, 1, 1, 36, 462, 2002, 3060, 1771, 406, 36, 1, 1, 45, 792, 5005, 11628, 10626, 4060, 666, 45, 1, 1, 55, 1287, 11440, 38760, 53130, 31465, 8436
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| The row sums form A098569: {1,2,5,14,43,143,510,1936,7775,32869,...}. How do the terms of row k tend to be distributed as k grows?
Remarkably, column k of the matrix inverse (A121434) equals signed column k of the triangular matrix power: A107876^(k*(k+1)/2) for k>=0. - Paul D. Hanna, Aug 25 2006
Surprisingly, the row sums (A098569) equal the row sums of triangle A131338. - Paul D. Hanna, Aug 30 2007
Number of sequences S = s(1)s(2)...s(n) such that S contains m 0s, for 1<=j<=n, s(j)<j and s(j-s(j)) = 0, for 1<j<=n, if s(j) positive, then s(j-1) < s(j). [Frank Ruskey, Apr 15 2011]
As a rectangular array read by antidiagonals R(n,k) (n>=2, k>=0) is the number of labeled graphs on n nodes that have exactly k arcs where multiple arcs are allowed to connect distinct vertex pairs. R(n,k) = C(C(n,2)+k-1,k). See example below. (*Geoffrey Critzer, Nov 12 2011*)
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REFERENCES
| Khamis, Soheir M., Height counting of unlabeled interval and N-free posets. Discrete Math. 275 (2004), no. 1-3, 165-175.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
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FORMULA
| T(n, k) = C( (k+1)*(k+2)/2 + n-k-1, n-k).
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EXAMPLE
| G.f. of columns are: 1/(1-x), 1/(1-x)^3, 1/(1-x)^6, 1/(1-x)^10, 1/(1-x)^15, ...
Rows begin:
[1],
[1,1],
[1,3,1],
[1,6,6,1],
[1,10,21,10,1],
[1,15,56,55,15,1],
[1,21,126,220,120,21,1],
[1,28,252,715,680,231,28,1],
[1,36,462,2002,3060,1771,406,36,1],
[1,45,792,5005,11628,10626,4060,666,45,1],
[1,55,1287,11440,38760,53130,31465,8436,1035,55,1],
[1,66,2002,24310,116280,230230,201376,82251,16215,1540,66,1],...
In reference to comment about s(1)s(2)...s(n) above,
a(4,2) = 6 = |{0012, 0013, 0023, 0101, 0103, 0120}| and
a(4,3) = 6 = |{0001, 0002, 0003, 0010, 0020, 0100}|. [Frank Ruskey, Apr 15 2011]
In reference to comment about multigraphs above,
1, 1, 1, 1, 1, 1, ... 2 nodes
1, 3, 6, 10, 15, 21, ... 3 nodes
1, 6, 21, 56, 126, 252, ... .
1, 10, 55, 220, 715, 2002, ... .
1, 15, 120, 680, 3060, 11628, ... .
1, 21, 231, 1771, 10626, 58130, ... .
(*Geoffrey Critzer, Nov 12 2011*)
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MATHEMATICA
| t[n_, k_] = Binomial[(k+1)*(k+2)/2 + n-k-1, n-k]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* From Jean-François Alcover, Jul 18 2011 *)
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PROG
| (PARI) T(n, k)=binomial((k+1)*(k+2)/2+n-k-1, n-k)
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CROSSREFS
| Cf. A098569.
Cf. A121434 (inverse); variants: A122175, A122176, A122177; A107876.
Cf. A131338.
Sequence in context: A107105 A088925 A100862 * A180959 A131235 A202812
Adjacent sequences: A098565 A098566 A098567 * A098569 A098570 A098571
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Sep 15 2004
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