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A239830 Triangular array:  T(n,k) = number of partitions of 2n that have alternating sum 2k, with T(0,0) = 1 for convenience. 37
1, 1, 1, 2, 2, 1, 3, 5, 2, 1, 5, 9, 5, 2, 1, 7, 17, 10, 5, 2, 1, 11, 28, 20, 10, 5, 2, 1, 15, 47, 35, 20, 10, 5, 2, 1, 22, 73, 62, 36, 20, 10, 5, 2, 1, 30, 114, 102, 65, 36, 20, 10, 5, 2, 1, 42, 170, 167, 109, 65, 36, 20, 10, 5, 2, 1, 56, 253, 262, 182, 110 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Suppose that p, with parts x(1) >= x(2) >= ... >= x(k), is a partition of n.  Define AS(p), the alternating sum of p, by x(1) - x(2) + x(3) - ... + ((-1)^(k-1))*x(k); note that AS(p) has the same parity as n.  Column 1 is given by T(n,1) = A000041(n) for n >= 0, which is the number of partitions of 2n having AS(p) = 0, for n >= 1.  Columns 2 and 3 are essentially A000567 and A000710, and the limiting column (after deleting initial 0's), A000712.  The sum of numbers in row n is A000041(2n).  The corresponding array for partitions into distinct parts is given by A152146 (defined as the number of unrestricted partitions of 2n into 2k even parts).

LINKS

Clark Kimberling and Alois P. Heinz, Rows n = 0..140, flattened (first 22 rows from Clark Kimberling)

EXAMPLE

First nine rows:

1

1 ... 1

2 ... 2 ... 1

3 ... 5 ... 2 ... 1

5 ... 9 ... 5 ... 2 ... 1

7 ... 17 .. 10 .. 5 ... 2 ... 1

11 .. 28 .. 20 .. 10 .. 5 ... 2 ... 1

15 .. 47 .. 35 .. 20 .. 10 .. 5 ... 2 ... 1

22 .. 73 .. 62 .. 36 .. 20 .. 10 .. 5 ... 2 ... 1

The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111, with respective alternating sums 6, 4, 2, 4, 0, 2, 2, 2, 0, 2, 0, so that row 3 (counting the top row as row 0) of the array is 3 .. 5 .. 2 .. 1.

MAPLE

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,

      expand(b(n, i-1, t)+`if`(i>n, 0, b(n-i, i, -t)*x^((t*i)/2)))))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(2*n$2, 1)):

seq(T(n), n=0..14);  # Alois P. Heinz, Mar 30 2014

MATHEMATICA

z = 16; s[w_] := s[w] = Total[Take[#, ;; ;; 2]] - Total[Take[Rest[#], ;; ;; 2]] &[w]; c[n_] := c[n] = Table[s[IntegerPartitions[n][[k]]], {k, 1, PartitionsP[n]}]; t[n_, k_] := Count[c[2 n], 2 k]; t[0, 0] = 1; u = Table[t[n, k], {n, 0, z}, {k, 0, n}]

TableForm[u]  (* A239830, array *)

Flatten[u]    (* A239830, sequence *)

(* Peter J. C. Moses, Mar 21 2014 *)

CROSSREFS

Cf. A239829, A239830, A239833.

Sequence in context: A165997 A046752 A086350 * A140767 A060850 A208336

Adjacent sequences:  A239827 A239828 A239829 * A239831 A239832 A239833

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Mar 28 2014

STATUS

approved

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Last modified May 26 10:17 EDT 2022. Contains 354086 sequences. (Running on oeis4.)