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A249543 Square array T(m,n) of integer partitions with m addends n+1, read by antidiagonals.

%I A249543 #26 Oct 29 2021 06:21:10
%S A249543 1,2,3,4,9,7,6,20,26,15,10,40,72,68,30,14,75,171,220,159,56,21,133,
%T A249543 379,614,603,352,101,29,229,786,1559,1928,1525,732,176,41,383,1568,
%U A249543 3700,5564,5534,3618,1465,297
%N A249543 Square array T(m,n) of integer partitions with m addends n+1, read by antidiagonals.
%C A249543 T(m,n) is the integer partition with m times the addend n+1 (and no other non-one addends) given as index number of A194602.
%C A249543 The entries in the array A249544 are also in the sequence A194602. This array T contains the index numbers of A194602 corresponding to the entries of that array: A194602(T(m,n)) = A249544(m,n).
%C A249543 Row 1 is A000065, column 1 is A058695 (both with shifted index).
%H A249543 Tilman Piesk, <a href="/A249543/b249543.txt">First 113 rows of the triangle, flattened</a>
%H A249543 Tilman Piesk, <a href="https://oeis.org/A194602/a194602.txt">Table for A194602</a>
%H A249543 Li-yao Xia, <a href="/A194602/a194602_1.txt">Identities for A194602</a>
%F A249543 A194602(T(m,n)) =  A249544(m,n).
%F A249543 T(1,n) = A000065(n+1) = p(n+1) - 1.
%F A249543 T(2,n) = p(2*(n+1)) - 2.
%F A249543 T(3,n) = p(3*(n+1)) - floor((n+1)/2) - 3.
%F A249543 T(m,1) = A058695(m-1) = p(2n-1).
%F A249543 p is the sequence of partition numbers A000041. (See "Identities for A194602" link.)
%e A249543 T(5,2) = 159.
%e A249543 A194602(159) = 14043. (So A249544(5,2) = 14043.)
%e A249543 14043 in binary is 11011011011011. That corresponds to the integer partition with 5 times the addend 3. (See row 159 in "Table for A194602" link.)
%e A249543 Array begins:
%e A249543    n    1    2    3    4    5    6    7   8   9
%e A249543 m
%e A249543 1       1    2    4    6   10   14   21  29  41
%e A249543 2       3    9   20   40   75  133  229 383
%e A249543 3       7   26   72  171  379  786 1568
%e A249543 4      15   68  220  614 1559 3700
%e A249543 5      30  159  603 1928 5564
%e A249543 6      56  352 1525 5534
%e A249543 7     101  732 3618
%e A249543 8     176 1465
%e A249543 9     297
%Y A249543 Cf. A194602, A249544, A000065, A058695, A000041.
%K A249543 nonn,tabl
%O A249543 1,2
%A A249543 _Tilman Piesk_, Oct 31 2014