A152146 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of 2n into 2k odd parts.

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, 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, 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

Offset

0,8

Comments

In both this and A152157, reading columns downwards "converges" to A000041.

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

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

T(n,k) = A152140(2n,2k).

Example

Triangle begins:
  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   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
  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
  0  7 27  35  29  22  15 11  7  5  3  2  1  1
  0  7 34  44  40  30  22 15 11  7  5  3  2  1 1
  0  8 39  58  52  42  30 22 15 11  7  5  3  2 1 1
  0  8 47  71  70  55  42 30 22 15 11  7  5  3 2 1 1
  0  9 54  90  89  75  56 42 30 22 15 11  7  5 3 2 1 1
  0  9 64 110 116  97  77 56 42 30 22 15 11  7 5 3 2 1 1
  0 10 72 136 146 128 100 77 56 42 30 22 15 11 7 5 3 2 1 1
From Gus Wiseman, Jun 20 2021: (Start)
For example, row n = 6 counts the following partitions (B = 11):
  (75)  (3333)  (333111)  (33111111)  (3111111111)  (111111111111)
  (93)  (5331)  (531111)  (51111111)
  (B1)  (5511)  (711111)
        (7311)
        (9111)
The corresponding strict partitions are:
  (7,5)      (8,4)      (9,3)    (10,2)   (11,1)  (12)
  (6,5,1)    (5,4,3)    (7,3,2)  (9,2,1)
  (5,4,2,1)  (6,4,2)    (8,3,1)
             (7,4,1)
             (6,3,2,1)
The corresponding normal partitions are:
  43221    33321     3321111    321111111   21111111111  111111111111
  322221   332211    32211111   2211111111
  2222211  432111    222111111
           3222111
           22221111
(End)

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-2)+`if`(i>n, 0, expand(sqrt(x)*b(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(2*n, 2*n-1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jun 21 2021

Mathematica

ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&ats[#]==k&]], {n, 0, 30, 2}, {k, 0, n, 2}] (* Gus Wiseman, Jun 20 2021 *)

Crossrefs

Cf. A035294 (row sums), A107379, A152140, A152157.

Column k = 1 is A004526.

Column k = 2-8 is A026810 - A026816.

The non-strict version is A239830.

The reverse non-strict version is A344610.

The reverse version is A344649

A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

A067659 counts strict partitions of odd length.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A124754 gives alternating sum of standard compositions (reverse: A344618).

A316524 is the alternating sum of the prime indices of n (reverse: A344616).

A344611 counts partitions of 2n with reverse-alternating sum >= 0.

Cf. A006330, A027187, A116406, A120452, A239829, A343941, A344607, A344608, A344609, A344650, A344651, A344739, A344741.

Sequence in context: A060582 A060450 A180918 * A025860 A322285 A152487

Adjacent sequences: A152143 A152144 A152145 * A152147 A152148 A152149

Keyword

nonn,tabl

Author

R. J. Mathar, Sep 25 2009, indices corrected Jul 09 2012

Status

approved