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A344651 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2. 37
1, 1, 1, 1, 2, 1, 2, 2, 1, 4, 2, 1, 3, 5, 2, 1, 7, 5, 2, 1, 5, 9, 5, 2, 1, 12, 10, 5, 2, 1, 7, 17, 10, 5, 2, 1, 19, 19, 10, 5, 2, 1, 11, 28, 20, 10, 5, 2, 1, 30, 33, 20, 10, 5, 2, 1, 15, 47, 35, 20, 10, 5, 2, 1, 45, 57, 36, 20, 10, 5, 2, 1, 22, 73, 62, 36, 20, 10, 5, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so T(n,k) is the number of integer partitions of n with k odd parts in the conjugate partition, which is also the number of partitions of n with k odd parts.

Also the number of integer partitions of n with odd-indexed parts (odd bisection) summing to k, ceiling(n/2) <= k <= n. The even-indexed version is A346633. - Gus Wiseman, Nov 29 2021

LINKS

Table of n, a(n) for n=0..80.

EXAMPLE

Triangle begins:

   1

   1

   1   1

   2   1

   2   2   1

   4   2   1

   3   5   2   1

   7   5   2   1

   5   9   5   2   1

  12  10   5   2   1

   7  17  10   5   2   1

  19  19  10   5   2   1

  11  28  20  10   5   2   1

  30  33  20  10   5   2   1

  15  47  35  20  10   5   2   1

  45  57  36  20  10   5   2   1

  22  73  62  36  20  10   5   2   1

  67  92  64  36  20  10   5   2   1

  30 114 102  65  36  20  10   5   2   1

  97 147 107  65  36  20  10   5   2   1

Row n = 10 counts the following partitions (A = 10):

  (55)          (64)         (73)       (82)     (91)   (A)

  (3322)        (442)        (433)      (622)    (811)

  (4411)        (541)        (532)      (721)

  (222211)      (3331)       (631)      (7111)

  (331111)      (4222)       (5221)     (61111)

  (22111111)    (4321)       (6211)

  (1111111111)  (5311)       (42211)

                (22222)      (52111)

                (32221)      (511111)

                (33211)      (4111111)

                (43111)

                (322111)

                (421111)

                (2221111)

                (3211111)

                (31111111)

                (211111111)

The conjugate version is:

  (A)      (55)      (3331)     (331111)    (31111111)   (1111111111)

  (64)     (73)      (5311)     (511111)    (211111111)

  (82)     (91)      (7111)     (3211111)

  (442)    (433)     (33211)    (4111111)

  (622)    (532)     (43111)    (22111111)

  (4222)   (541)     (52111)

  (22222)  (631)     (61111)

           (721)     (322111)

           (811)     (421111)

           (3322)    (2221111)

           (4321)

           (4411)

           (5221)

           (6211)

           (32221)

           (42211)

           (222211)

MATHEMATICA

ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];

Table[Length[Select[IntegerPartitions[n], ats[#]==k&]], {n, 0, 15}, {k, Mod[n, 2], n, 2}]

CROSSREFS

This is A103919 with all zeros removed.

The strict version is A152146 interleaved with A152157.

The rows are those of A239830 interleaved with those of A239829.

The reverse version is the right half of A344612.

The strict reverse version is the right half of A344739.

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

A027187 counts partitions with rev-alternating sum <= 0, ranked by A028260.

A124754 lists alternating sums of standard compositions (reverse: A344618).

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

A325534/A325535 count separable/inseparable partitions.

A344607 counts partitions with rev-alternating sum >= 0, ranked by A344609.

A344608 counts partitions with rev-alternating sum < 0, ranked by A119899.

A344610 counts partitions of n by positive rev-alternating sum.

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

A345197 counts compositions by sum, length, and alternating sum.

A346697 gives the sum of odd-indexed prime indices (reverse: A346699).

A346702 represents the odd bisection of compositions, sums A209281.

Cf. A000070, A000097, A003242, A006330, A025047, A097805, A114121, A116406, A131577, A344617, A344649, A344650, A344654, A346633.

Sequence in context: A177954 A337312 A077053 * A308558 A105272 A060438

Adjacent sequences:  A344648 A344649 A344650 * A344652 A344653 A344654

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Jun 05 2021

STATUS

approved

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