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Search: id:A049597
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| A049597 |
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Triangular array T(n,k) in which k-th column gives coefficients of sum of Gaussian polynomials [k,m] for m=0..k. |
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+0
5
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| 1, 0, 2, 0, 0, 3, 0, 0, 1, 4, 0, 0, 0, 2, 5, 0, 0, 0, 2, 3, 6, 0, 0, 0, 0, 4, 4, 7, 0, 0, 0, 0, 3, 6, 5, 8, 0, 0, 0, 0, 1, 6, 8, 6, 9, 0, 0, 0, 0, 0, 6, 9, 10, 7, 10, 0, 0, 0, 0, 0, 2, 11, 12, 12, 8, 11, 0, 0, 0, 0, 0, 2, 9, 16, 15, 14, 9, 12, 0, 0, 0, 0, 0, 0, 7, 16, 21, 18, 16, 10, 13, 0, 0, 0, 0, 0, 0
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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It appears that T(n-1,k-1) is the number of partitions of n with k objects in the first hook; i.e., with (largest part size) + (number of parts) - 1 = k. If this is correct, we have T(n-1,k-1) = sum_{j<=min(k,n-k-2)} (k-j) * T(k-1,j-1) with T(n-1,n-1) = n. Equivalently, T(n-1,k-1) = T(n-2,k-2) + sum(j<=min(k,n-k-2)} T(k-1,j-1), and thus T(n-1,k-1) = 2*T(n-2,k-2) - T(n-3,k-3) + T(k-1,n-k-3). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 27 2008
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REFERENCES
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Cf. G. E. Andrews, Theory of Partitions, 1976, pages 240-243
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EXAMPLE
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Table begins:
1
0 2
0 0 3
0 0 1 4
0 0 0 2 5
0 0 0 2 3 6
0 0 0 0 4 4 7
0 0 0 0 3 6 5 8
For k=4 the 5 polynomials have coefficients 1; 1 1 1 1; 1 1 2 1 1; 1 1 1 1; 1; which sum to 5 3 4 3 1, giving column 4.
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MAPLE
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a := n->sort(simplify(sum(product((1-q^i), i=n-r+1..n)/product((1-q^j), j=1..r), r=0..n))):T := proc(n, k) if k=n then n+1 elif k>n then 0 else coeff(a(k), q^(n-k)) fi end:seq(seq(T(n, k), k=0..n), n=0..21);
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CROSSREFS
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The nonzero entries of the columns are the rows of A083906.
Sequence in context: A127648 A132681 A132825 this_sequence A035377 A136274 A114699
Adjacent sequences: A049594 A049595 A049596 this_sequence A049598 A049599 A049600
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KEYWORD
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nonn,tabl
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AUTHOR
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Alford Arnold (Alford1940(AT)aol.com)
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2004
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