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%I #24 Feb 28 2024 11:44:05
%S 1,0,1,0,1,1,0,2,1,1,0,2,2,1,1,0,3,3,2,1,1,0,3,5,3,2,1,1,0,4,6,5,3,2,
%T 1,1,0,4,9,7,5,3,2,1,1,0,5,11,11,7,5,3,2,1,1,0,5,15,14,11,7,5,3,2,1,1,
%U 0,6,18,20,15,11,7,5,3,2,1,1,0,6,23,26,22,15,11,7,5,3,2,1,1
%N Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of 2n into 2k odd parts.
%C In both this and A152157, reading columns downwards "converges" to A000041.
%C Also the number of strict integer partitions of 2n with alternating sum 2k. Also the number of normal integer partitions of 2n of which 2k parts are odd, where a partition is normal if it covers an initial interval of positive integers. - _Gus Wiseman_, Jun 20 2021
%H Alois P. Heinz, <a href="/A152146/b152146.txt">Rows n = 0..200, flattened</a>
%F T(n,k) = A152140(2n,2k).
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 2 1 1
%e 0 2 2 1 1
%e 0 3 3 2 1 1
%e 0 3 5 3 2 1 1
%e 0 4 6 5 3 2 1 1
%e 0 4 9 7 5 3 2 1 1
%e 0 5 11 11 7 5 3 2 1 1
%e 0 5 15 14 11 7 5 3 2 1 1
%e 0 6 18 20 15 11 7 5 3 2 1 1
%e 0 6 23 26 22 15 11 7 5 3 2 1 1
%e 0 7 27 35 29 22 15 11 7 5 3 2 1 1
%e 0 7 34 44 40 30 22 15 11 7 5 3 2 1 1
%e 0 8 39 58 52 42 30 22 15 11 7 5 3 2 1 1
%e 0 8 47 71 70 55 42 30 22 15 11 7 5 3 2 1 1
%e 0 9 54 90 89 75 56 42 30 22 15 11 7 5 3 2 1 1
%e 0 9 64 110 116 97 77 56 42 30 22 15 11 7 5 3 2 1 1
%e 0 10 72 136 146 128 100 77 56 42 30 22 15 11 7 5 3 2 1 1
%e From _Gus Wiseman_, Jun 20 2021: (Start)
%e For example, row n = 6 counts the following partitions (B = 11):
%e (75) (3333) (333111) (33111111) (3111111111) (111111111111)
%e (93) (5331) (531111) (51111111)
%e (B1) (5511) (711111)
%e (7311)
%e (9111)
%e The corresponding strict partitions are:
%e (7,5) (8,4) (9,3) (10,2) (11,1) (12)
%e (6,5,1) (5,4,3) (7,3,2) (9,2,1)
%e (5,4,2,1) (6,4,2) (8,3,1)
%e (7,4,1)
%e (6,3,2,1)
%e The corresponding normal partitions are:
%e 43221 33321 3321111 321111111 21111111111 111111111111
%e 322221 332211 32211111 2211111111
%e 2222211 432111 222111111
%e 3222111
%e 22221111
%e (End)
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-2)+`if`(i>n, 0, expand(sqrt(x)*b(n-i, i)))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(2*n, 2*n-1)):
%p seq(T(n), n=0..12); # _Alois P. Heinz_, Jun 21 2021
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&ats[#]==k&]],{n,0,30,2},{k,0,n,2}] (* _Gus Wiseman_, Jun 20 2021 *)
%Y Cf. A035294 (row sums), A107379, A152140, A152157.
%Y Column k = 1 is A004526.
%Y Column k = 2-8 is A026810 - A026816.
%Y The non-strict version is A239830.
%Y The reverse non-strict version is A344610.
%Y The reverse version is A344649
%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A067659 counts strict partitions of odd length.
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A124754 gives alternating sum of standard compositions (reverse: A344618).
%Y A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y Cf. A006330, A027187, A116406, A120452, A239829, A343941, A344607, A344608, A344609, A344650, A344651, A344739, A344741.
%K nonn,tabl
%O 0,8
%A _R. J. Mathar_, Sep 25 2009, indices corrected Jul 09 2012
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