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

%I A260672 #23 Feb 16 2025 08:33:26
%S A260672 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 A260672 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 A260672 2,15,14,13,10,8,3,0,16,15,14,11,9,4,1,17,16
%N A260672 Table read by rows: T(n,k) = n - A001318(k), k = 0 .. A193832(n)-1.
%C A260672 Column k starts at row A001318(k); each column = A001477.
%H A260672 Reinhard Zumkeller, <a href="/A260672/b260672.txt">Rows n = 0..1000 of triangle, flattened</a>
%H A260672 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 A260672 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%H A260672 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentagonal_number_theorem">Pentagonal number theorem</a>
%F A260672 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 A260672 .   0:    0
%e A260672 .   1:    1   0
%e A260672 .   2:    2   1   0
%e A260672 .   3:    3   2   1
%e A260672 .   4:    4   3   2
%e A260672 .   5:    5   4   3   0
%e A260672 .   6:    6   5   4   1
%e A260672 .   7:    7   6   5   2   0
%e A260672 .   8:    8   7   6   3   1
%e A260672 .   9:    9   8   7   4   2
%e A260672 .  10:   10   9   8   5   3
%e A260672 .  11:   11  10   9   6   4
%e A260672 .  12:   12  11  10   7   5   0
%e A260672 .  13:   13  12  11   8   6   1
%e A260672 .  14:   14  13  12   9   7   2
%e A260672 .  15:   15  14  13  10   8   3   0
%e A260672 .  16:   16  15  14  11   9   4   1
%e A260672 .  17:   17  16  15  12  10   5   2
%e A260672 .  18:   18  17  16  13  11   6   3
%e A260672 .  19:   19  18  17  14  12   7   4
%e A260672 .  20:   20  19  18  15  13   8   5  .
%o A260672 (Haskell)
%o A260672 a260672 n k = a260672_tabf !! n !! k
%o A260672 a260672_row n = a260672_tabf !! n
%o A260672 a260672_tabf = map (takeWhile (>= 0) . flip map a001318_list . (-)) [0..]
%Y A260672 Cf. A001318, A193832 (row lengths), A000041, A087960, A054440, A260664, A260706 (row sums).
%K A260672 nonn,tabf,look
%O A260672 0,4
%A A260672 _Reinhard Zumkeller_, Nov 15 2015