A152146 -id:A152146 - cp's OEIS frontend

A343941 Number of strict integer partitions of 2n with reverse-alternating sum 4.

0, 0, 1, 0, 1, 2, 3, 3, 4, 5, 7, 8, 10, 11, 14, 15, 18, 20, 23, 25, 29, 31, 35, 38, 42, 45, 50, 53, 58, 62, 67, 71, 77, 81, 87, 92, 98, 103, 110, 115, 122, 128, 135, 141, 149, 155, 163, 170, 178, 185, 194, 201, 210, 218, 227, 235, 245, 253, 263, 272, 282, 291, 302

Offset: 0

Gus Wiseman, Jun 09 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts, so a(n) is the number of strict odd-length integer partitions of 2n whose conjugate has exactly 4 odd parts (first example). By conjugation, this is also the number partitions of 2n covering an initial interval and containing exactly four odd parts, one of which is the greatest (second example).

			The a(2) = 1 through a(12) = 10 strict partitions (empty column indicated by dot, A..D = 10..13):
  4   .  521   532   543   653   763     873     983     A93     BA3
               631   642   752   862     972     A82     B92     CA2
                     741   851   961     A71     B81     C91     DA1
                                 64321   65421   65432   76432   76542
                                         75321   75431   76531   86541
                                                 76421   86431   87432
                                                 86321   87421   87531
                                                         97321   97431
                                                                 98421
                                                                 A8321
The a(2) = 1 through a(8) = 5 partitions covering an initial interval:
  1111  .  32111   33211    33321     333221     543211      543321
                   322111   332211    3322211    3332221     5432211
                            3222111   32222111   33222211    33322221
                                                 322222111   332222211
                                                             3222222111

The non-reverse non-strict version is A000710.
The non-reverse version is A026810.
The non-strict version is column k = 2 of A344610.
This is column k = 2 of A344649.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums 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. A000070, A000097, A003242, A006330, A027187, A119899, A152146, A239830, A325535, A344604, A344607, A344608, A344650, A344739.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==4&]],{n,0,30,2}]

More terms from Bert Dobbelaere, Jun 12 2021

A152157 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of 2n+1 into 2k+1 odd parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 3, 2, 1, 1, 1, 5, 5, 3, 2, 1, 1, 1, 7, 7, 5, 3, 2, 1, 1, 1, 8, 10, 7, 5, 3, 2, 1, 1, 1, 10, 13, 11, 7, 5, 3, 2, 1, 1, 1, 12, 18, 15, 11, 7, 5, 3, 2, 1, 1, 1, 14, 23, 21, 15, 11, 7, 5, 3, 2, 1, 1, 1, 16, 30, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 0

R. J. Mathar, Sep 25 2009

This number triangle satisfies T(n,k) = A008284(n+k+1,2*k+1), n,k >= 0. This means that T(n,k) is also the number of partitions of N:=n+k+1 into M:=2*k+1 parts. For the proof add 1 to every odd part of each partition of n':=2*n+1 into m:=2*k+1 parts which are all odd, and divide each part by a factor of 2, thus obtaining a partition of n+k+1 into m=2*k+1 parts. All partitions of N,for N>=1, into an odd number of parts M (M from {1,...,N}) are reached: just take k=(M-1)/2 and n=N-1-k. Each partition of N into an odd number of parts can only arise once from the given recipe (for given N and M the k and n values are unique). See also a comment by Franklin T. Adams-Watters on A152140. - Wolfdieter Lang, Jul 09 2012

			Triangle begins:
1
1  1
1  1   1
1  2   1   1
1  3   2   1   1
1  4   3   2   1   1
1  5   5   3   2   1   1
1  7   7   5   3   2   1  1
1  8  10   7   5   3   2  1  1
1 10  13  11   7   5   3  2  1  1
1 12  18  15  11   7   5  3  2  1  1
1 14  23  21  15  11   7  5  3  2  1  1
1 16  30  28  22  15  11  7  5  3  2  1  1
1 19  37  38  30  22  15 11  7  5  3  2  1  1
1 21  47  49  41  30  22 15 11  7  5  3  2  1 1
1 24  57  65  54  42  30 22 15 11  7  5  3  2 1 1
1 27  70  82  73  56  42 30 22 15 11  7  5  3 2 1  1
1 30  84 105  94  76  56 42 30 22 15 11  7  5 3 2  1 1
1 33 101 131 123  99  77 56 42 30 22 15 11  7 5 3  2 1 1
1 37 119 164 157 131 101 77 56 42 30 22 15 11 7 5  3 2 1 1
From _Wolfdieter Lang_, Jul 09 2012 (Start)
T(5,1) = 4 from the four partitions of 11 into 3 parts, all of which are odd: [1,1,9], [1,3,7], [1,5,5] and [3,3,5].
T(5,1) = 4 from the four partitions of 7 = 5+1+1 into 3 parts:
[1,1,5], [1,2,4], [1,3,3] and [2,2,3].
(End)

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

Cf. A078408 (row sums), A107379, A152140, A152146, A008284.

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

(* p = A008284 *) p[n_, 1] = 1; p[n_, k_] := p[n, k] = If[n >= k, Sum[p[n - i, k - 1], {i, 1, n - 1}] - Sum[p[n - i, k], {i, 1, k - 1}], 0];
T[n_, k_] := p[n + k + 1, 2 k + 1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 28 2019, after Wolfdieter Lang *)

T(n,k) = A152140(2n+1,2k+1).
T(n,k) = p(n+k+1,2*k+1), n >= 0, k >= 0, with p(N,M)= A008284(N,M), the number of partitions of N into M parts. See the sketch of the proof given above as a comment. - Wolfdieter Lang, Jul 09 2012
O.g.f. for column k: (x^k)/product(1-x^j,j=1..(2*k+1)), k>=0.
  From the o.g.f.s of A008284. - Wolfdieter Lang, Jul 10 2012

Indices corrected by R. J. Mathar, Jul 09 2012

A152140 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n into k odd parts.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 3, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1

Offset: 0

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

The number of partitions of n into k odd parts is equal to the number of partitions of (n+k)/2 into k parts; or equivalently the number of partitions of (n-k)/2 into at most k parts. - Franklin T. Adams-Watters, Sep 25 2009

			n= 0, k= 0: [];
n= 1, k= 1: [1] ;
n= 2, k= 2: [1, 1] ;
n= 3, k= 1: [3] ;
n= 3, k= 3: [1, 1, 1] ;
n= 4, k= 2: [1, 3] ;
n= 4, k= 4: [1, 1, 1, 1];
n= 5, k= 1: [5];
n= 5, k= 3: [1, 1, 3];
n= 5, k= 5: [1, 1, 1, 1, 1];
n= 6, k= 2: [3, 3] or [1, 5];
n= 6, k= 4: [1, 1, 1, 3];
n= 6, k= 6: [1, 1, 1, 1, 1, 1];
Triangle begins:
1
0 1
0 0 1
0 1 0 1
0 0 1 0 1
0 1 0 1 0 1
0 0 2 0 1 0 1
0 1 0 2 0 1 0 1
0 0 2 0 2 0 1 0 1
0 1 0 3 0 2 0 1 0 1
0 0 3 0 3 0 2 0 1 0 1
0 1 0 4 0 3 0 2 0 1 0 1
0 0 3 0 5 0 3 0 2 0 1 0 1
0 1 0 5 0 5 0 3 0 2 0 1 0 1
0 0 4 0 6 0 5 0 3 0 2 0 1 0 1
0 1 0 7 0 7 0 5 0 3 0 2 0 1 0 1
0 0 4 0 9 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 8 0 10 0 7 0 5 0 3 0 2 0 1 0 1
0 0 5 0 11 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 10 0 13 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 5 0 15 0 14 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 12 0 18 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 6 0 18 0 20 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 14 0 23 0 21 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 6 0 23 0 26 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 16 0 30 0 28 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 7 0 27 0 35 0 29 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 19 0 37 0 38 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 7 0 34 0 44 0 40 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 21 0 47 0 49 0 41 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 8 0 39 0 58 0 52 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 24 0 57 0 65 0 54 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 8 0 47 0 71 0 70 0 55 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 27 0 70 0 82 0 73 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 9 0 54 0 90 0 89 0 75 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 30 0 84 0 105 0 94 0 76 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000009 (row sums), A097304, A107379, A152146, A152157.

b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 elif irem(i, 2)=0 then b(n, i-1)
      else []; for j from 0 to n/i do zip((x, y)->x+y, %,
      [0$j, b(n-i*j, i-2)], 0) od; %[] fi
    end:
T:= n-> b(n$2):
seq(T(n), n=0..13);  # Alois P. Heinz, May 31 2013

nn = 10; CoefficientList[
Series[Product[1/(1 - y x^i), {i, 1, nn, 2}], {x, 0, nn}], {x, y}] (* Geoffrey Critzer, May 31 2013 *)

A318156 Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k(2k-1)) / Product_{j=1..2*k-1} (1 - x^j).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 35, 44, 55, 69, 85, 104, 127, 154, 186, 224, 268, 320, 381, 452, 534, 630, 741, 869, 1017, 1187, 1382, 1606, 1862, 2155, 2489, 2869, 3301, 3792, 4349, 4979, 5692, 6497, 7405, 8429, 9581, 10876, 12331, 13963, 15792, 17840, 20131, 22691

Offset: 0

Ilya Gutkovskiy, Aug 19 2018

nonn

Partial sums of A067659.

			From _Gus Wiseman_, Jul 18 2021: (Start)
Also the number of strict integer partitions of 2n+1 of even length with exactly one odd part. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (2,1)  (3,2)  (4,3)  (5,4)  (6,5)   (7,6)      (8,7)      (9,8)
         (4,1)  (5,2)  (6,3)  (7,4)   (8,5)      (9,6)      (10,7)
                (6,1)  (7,2)  (8,3)   (9,4)      (10,5)     (11,6)
                       (8,1)  (9,2)   (10,3)     (11,4)     (12,5)
                              (10,1)  (11,2)     (12,3)     (13,4)
                                      (12,1)     (13,2)     (14,3)
                                      (6,4,2,1)  (14,1)     (15,2)
                                                 (6,4,3,2)  (16,1)
                                                 (8,4,2,1)  (6,5,4,2)
                                                            (8,4,3,2)
                                                            (8,6,2,1)
                                                            (10,4,2,1)
Also the number of integer partitions of 2n+1 covering an initial interval and having even maximum and alternating sum 1.
(End)

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.2 "Partitions into distinct parts", page 350.
Index entries for sequences related to partitions

Partial sums of A067659.
The following relate to strict integer partitions of 2n+1 of even length with exactly one odd part.
- Allowing any length gives A036469.
- The non-strict version is A306145.
- The version for odd length is A318155 (non-strict: A304620).
- Allowing any number of odd parts gives A343942 (odd bisection of A067661).
A000041 counts partitions.
A027187 counts partitions of even length (strict: A067661).
A078408 counts strict partitions of 2n+1 (odd bisection of A000009).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
Cf. A000070, A030229, A035294, A058696, A078616, A087447, A152146, A236559, A343941, A344611, A344739.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);

nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k - 1))/Product[(1 - x^j), {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] - QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman, Jul 18 2021 *)

a(n) = A036469(n) - A318155(n).
a(n) = A318155(n) - A078616(n).
a(n) ~ exp(Pi*sqrt(n/3)) * 3^(1/4) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, Aug 20 2018

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A343941 Number of strict integer partitions of 2n with reverse-alternating sum 4.

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A152157 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of 2n+1 into 2k+1 odd parts.

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A152140 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n into k odd parts.

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A318156 Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k*(2*k-1)) / Product_{j=1..2*k-1} (1 - x^j).

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A318156 Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k(2k-1)) / Product_{j=1..2*k-1} (1 - x^j).