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A035002
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Square array a(m,n) read by antidiagonals, where a(m,n) = sum(a(m-k,n), k=1..m-1)+sum(a(m,n-k), k=1..n-1).
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8
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1, 1, 1, 2, 2, 2, 4, 5, 5, 4, 8, 12, 14, 12, 8, 16, 28, 37, 37, 28, 16, 32, 64, 94, 106, 94, 64, 32, 64, 144, 232, 289, 289, 232, 144, 64, 128, 320, 560, 760, 838, 760, 560, 320, 128, 256, 704, 1328, 1944, 2329, 2329, 1944, 1328, 704, 256, 512, 1536, 3104, 4864, 6266
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| a(m,n) is the sum of all the entries above it plus the sum of all the entries to the left of it.
a(m,n) equals the number of ways to move a chess rook from the lower left corner to square (m,n), with the rook moving only up or right. - Francisco Santos (santosf(AT)unican.es), Oct 20 2005
a(m,n) is the number of Nim games that start with two piles of stones of sizes m and n. [From Martin J. Erickson (erickson(AT)truman.edu), Dec 05 2008]
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REFERENCES
| C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.
M. Erickson, S. Fernando, K. Tran, Enumerating Rook and Queen Paths, Bulletin of the Institute for Combinatorics and Its Applications, Volume 60 (2010), 37--48 [From Martin J. Erickson (erickson(AT)truman.edu), Oct 21 2010]
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FORMULA
| G.f. A(n; x) for n-th row satisfies A(n; x) = Sum_{k=1..n} (1+x^k)*A(n-k; x), A(0; x) = 1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 03 2002
a(m+1, n+1)=2a(m+1, n)+2a(m, n+1)-3a(m, n); a(n, 1)=a(1, n)= A011782(n) - Francisco Santos (santosf(AT)unican.es), Oct 20 2005
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EXAMPLE
| Table begins:
1 1 2 4 8 16 32 64 ...
1 2 5 12 28 64 144 320 ...
2 5 14 37 94 232 560 1328 ...
4 12 37 106 289 760 1944 4864 ...
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MATHEMATICA
| a[n_, 1] = 2^(n-2); a[1, n_] = 2^(n-2); a[1, 1] = 1; a[m_, n_] := a[m, n] = Sum[a[m-k, n], {k, 1, m-1}] + Sum[a[m, n-k], {k, 1, n-1}]; Flatten[Table[a[m-n+1 , n], {m, 1, 11}, {n, 1, m}] ](* From Jean-François Alcover, Nov 04 2011 *)
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CROSSREFS
| Cf. A035001, A051708.
Row sums give A025192.
Sequence in context: A024681 A007495 A122385 * A032578 A035659 A008282
Adjacent sequences: A034999 A035000 A035001 * A035003 A035004 A035005
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| Erich Friedman (erich.friedman(AT)stetson.edu)
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