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Search: id:A064391
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| A064391 |
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Triangle T(n,k) with zero-th row {1} and row n for n >= 1 giving number of partitions of n with crank k, for -n <= k <= n. |
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+0
5
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| 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 2, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2
(list; graph; listen)
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OFFSET
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0,56
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COMMENT
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For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).
n-th row contains 2n+1 terms.
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REFERENCES
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G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
F. Garvan, New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc., 305 (1988), 47-77.
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FORMULA
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G.f. for k-th column is Sum((-1)^m*x^(k*m)*(x^((m^2+m)/2)-x^((m^2-m)/2)), m=1..infinity)/Product(1-x^m, k=1..infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 22 2004
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EXAMPLE
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1; 1,0,0; 1,0,0,0,1; 1,0,0,1,0,0,1; ...
{T(20, k), -20 <= k <=20} = {1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 19, 20, 26, 28, 34, 34, 39, 38, 41, 38, 39, 34, 34, 28, 26, 20, 19, 13, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1}.
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CROSSREFS
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Cf. A001522, A064410, A064428.
Adjacent sequences: A064388 A064389 A064390 this_sequence A064392 A064393 A064394
Sequence in context: A136571 A108149 A128583 this_sequence A086011 A124760 A077619
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KEYWORD
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nonn,tabf,nice,easy
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AUTHOR
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njas, Sep 29 2001
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 29 2001
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