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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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13
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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
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OFFSET
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0,5
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COMMENTS
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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
Number of sequences S = s(1)s(2)...s(n) such that S contains m 0's, 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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LINKS
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FORMULA
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T(n, k) = binomial((k+1)*(k+2)/2 + n-k-1, n-k).
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EXAMPLE
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G.f.s of columns: 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}|. (End)
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, ... . (End)
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MATHEMATICA
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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}]] (* Jean-François Alcover, Jul 18 2011 *)
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PROG
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(PARI) {T(n, k)=binomial((k+1)*(k+2)/2+n-k-1, n-k)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
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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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