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A064580
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Triangle associated with rooted trees with a degree constraint (A036765).
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6
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1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 9, 14, 13, 1, 5, 14, 28, 40, 36, 1, 6, 20, 48, 87, 118, 104, 1, 7, 27, 75, 161, 273, 357, 309, 1, 8, 35, 110, 270, 536, 866, 1100, 939, 1, 9, 44, 154, 423, 951, 1782, 2772, 3441, 2905, 1, 10, 54, 208, 630, 1572, 3310, 5928, 8946, 10900, 9118
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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Row sums = A161898: (1, 2, 5, 14, 41, 124, 384,...). - Gary W. Adamson, Jun 21 2009
Main diagonal is A036765. - Paul D. Hanna, Nov 18 2016
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LINKS
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Table of n, a(n) for n=0..65.
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FORMULA
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a(n, k) = a(n-1, k)+a(n-1, k-1)+a(n-1, k-2)+a(n-1, k-3) with a(0, 0)=1 and a(n, k)=0 if n<k or k<0.
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EXAMPLE
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Triangle begins:
1,
1, 1,
1, 2, 2,
1, 3, 5, 5,
1, 4, 9, 14, 13,
1, 5, 14, 28, 40, 36,
...
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MATHEMATICA
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a[n_, k_] /; 0 <= k <= n = a[n, k] = a[n - 1, k] + a[n - 1, k - 1] + a[n - 1, k - 2] + a[n - 1, k - 3]; a[0, 0] = 1; a[_, _] = 0;
Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018 *}
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PROG
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(Sage) # Function riordan_array defined in A256893.
M = riordan_array(1, x/(1+x+x^2+x^3), 12).inverse()
for m in M[1:]:
print [r for r in reversed(list(m)) if r <> 0] # Peter Luschny, Aug 17 2016
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CROSSREFS
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Columns include A000012, A000027, A000096. Main diagonal is A036765.
The sequence of triangles A010054 (Triangle Indicator), A007318 (Pascal), A026300 (Motzkin), A064580, ... converges to the triangle A009766 (Catalan).
Cf. A036765 (diagonal), A161898 (row sums).
Sequence in context: A139687 A188181 A064581 * A009766 A059718 A076038
Adjacent sequences: A064577 A064578 A064579 * A064581 A064582 A064583
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KEYWORD
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nonn,tabl
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AUTHOR
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Henry Bottomley, Sep 21 2001
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EXTENSIONS
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Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007
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STATUS
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approved
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