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A106597
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Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(n,0)=T(n,n)=1, T(n,k) = T(n-1,k-1) + T(n-1,k) + Sum_{i >= 1} T(n-2i,k-i).
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2
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1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 27, 27, 9, 1, 1, 11, 44, 72, 44, 11, 1, 1, 13, 65, 149, 149, 65, 13, 1, 1, 15, 90, 266, 388, 266, 90, 15, 1, 1, 17, 119, 431, 836, 836, 431, 119, 17, 1, 1, 19, 152, 652, 1585, 2150, 1585, 652, 152, 19, 1, 1, 21, 189, 937
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Next term is sum of two terms above you in previous row (as in Pascal's triangle A007318) plus sum of terms directly above you on a vertical line.
T(n,n-k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), and (s,s) for s>=1. [Joerg Arndt, Jul 01 2011]
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EXAMPLE
| Triangle begins:
1;
1, 1;
1, 3, 1;
1, 5, 5, 1;
1, 7, 14, 7, 1;
1, 9, 27, 27, 9, 1;
1, 11, 44, 72, 44, 11, 1;
1, 13, 65, 149, 149, 65, 13, 1;
1, 15, 90, 266, 388, 266, 90, 15, 1;
1, 17, 119, 431, 836, 836, 431, 119, 17, 1;
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PROG
| (PARI) /* same as in A092566, but last line (output) replaced by the following */
/* show as triangle T(n-k, k): */
{ for(n=0, N-1, for(k=0, n, print1(T(n-k, k), ", "); ); print(); ); }
/* Joerg Arndt, Jul 01 2011 */
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CROSSREFS
| Sequence in context: A026714 A008288 A144461 * A108359 A100936 A086620
Adjacent sequences: A106594 A106595 A106596 * A106598 A106599 A106600
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KEYWORD
| nonn,tabl,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), May 30 2005
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EXTENSIONS
| More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 10 2006
Definition corrected by Emilie Hogan, Oct 15 2009
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