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A260672 Table read by rows: T(n,k) = n - A001318(k), k = 0 .. A193832(n)-1. 6

%I

%S 0,1,0,2,1,0,3,2,1,4,3,2,5,4,3,0,6,5,4,1,7,6,5,2,0,8,7,6,3,1,9,8,7,4,

%T 2,10,9,8,5,3,11,10,9,6,4,12,11,10,7,5,0,13,12,11,8,6,1,14,13,12,9,7,

%U 2,15,14,13,10,8,3,0,16,15,14,11,9,4,1,17,16

%N Table read by rows: T(n,k) = n - A001318(k), k = 0 .. A193832(n)-1.

%C Column k starts at row A001318(k); each column = A001477.

%H Reinhard Zumkeller, <a href="/A260672/b260672.txt">Rows n = 0..1000 of triangle, flattened</a>

%H Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, <a href="https://doi.org/10.1006/jcta.1997.2846">A pentagonal number sieve</a>, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentagonal_number_theorem">Pentagonal number theorem</a>

%F Number of m-tuples of partitions of n that have no part in common = Sum(A087960(k)*A000041(T(n,k))^m: k = 0 .. A193832(n+1)-1), e.g. A054440 (m=2) and A260664 (m=3); see Wilf link: p. 2, (3).

%e . 0: 0

%e . 1: 1 0

%e . 2: 2 1 0

%e . 3: 3 2 1

%e . 4: 4 3 2

%e . 5: 5 4 3 0

%e . 6: 6 5 4 1

%e . 7: 7 6 5 2 0

%e . 8: 8 7 6 3 1

%e . 9: 9 8 7 4 2

%e . 10: 10 9 8 5 3

%e . 11: 11 10 9 6 4

%e . 12: 12 11 10 7 5 0

%e . 13: 13 12 11 8 6 1

%e . 14: 14 13 12 9 7 2

%e . 15: 15 14 13 10 8 3 0

%e . 16: 16 15 14 11 9 4 1

%e . 17: 17 16 15 12 10 5 2

%e . 18: 18 17 16 13 11 6 3

%e . 19: 19 18 17 14 12 7 4

%e . 20: 20 19 18 15 13 8 5 .

%o (Haskell)

%o a260672 n k = a260672_tabf !! n !! k

%o a260672_row n = a260672_tabf !! n

%o a260672_tabf = map (takeWhile (>= 0) . flip map a001318_list . (-)) [0..]

%Y Cf. A001318, A193832 (row lengths), A000041, A087960, A054440, A260664, A260706 (row sums).

%K nonn,tabf,look

%O 0,4

%A _Reinhard Zumkeller_, Nov 15 2015

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Last modified April 13 12:29 EDT 2021. Contains 342936 sequences. (Running on oeis4.)