A077028 -id:A077028 - cp's OEIS frontend

A224937 Number of partitions of n having T(n,k) odd parts in excess on even places over odd places.

0, 1, 1, 0, 0, 2, 0, 2, 0, 1, 0, 0, 5, 0, 0, 5, 0, 2, 1, 0, 10, 0, 0, 10, 0, 5, 2, 0, 20, 0, 0, 20, 0, 10, 0, 5, 0, 36, 0, 1, 0, 0, 36, 0, 20, 0, 0, 10, 0, 65, 0, 2, 0, 0, 65, 0, 36, 0, 0, 20, 0, 110, 0, 5, 1, 0, 110, 0, 65, 0, 0, 36, 0, 185, 0, 10, 2, 0, 185, 0, 110, 0, 0, 65, 0, 300, 0, 20

Offset: 0

Wouter Meeussen, Apr 20 2013

Row lengths are 2*floor((3 + sqrt(1+8*n))/4), k runs from -floor((3 + sqrt(1+8*n))/4) up to floor((-1 + sqrt(1+8*n))/4); row sums are A000041.

P. D. Hanna remarks that "zig-zag" diagonals/antidiagonals produce A077028 (Rascal triangle).

			In the table below, replace each integer i with A000720(i) to get the  current sequence:
-3     -2      -1       0       1       2 (= k)(n= )
                0       1                         0
                1       0                         1
                0       2                         2
        0       2       0       1                 3
        0       0       3       0                 4
        0       3       0       2                 5
        1       0       4       0                 6
        0       4       0       3                 7
        2       0       5       0                 8
        0       5       0       4                 9
0       3       0       6       0       1         10
0       0       6       0       5       0         11
0       4       0       7       0       2         12
0       0       7       0       6       0         13
0       5       0       8       0       3         14
1       0       8       0       7       0         15
...
The table then starts as:
0  0,1
1  1,0
2  0,2
3  0,2,0,1
4  0,0,5,0
5  0,5,0,2
6  1,0,10,0
7  0,10,0,5
8  2,0,20,0
9  0,20,0,10
10 0,5,0,36,0,1
  ...
The partitions of n=5 then give (0,5,0,2) for k=(-2,-1,0,1); this corresponds to 5 partitions with -1 excess odd parts on even over odd positions, and 2 with 1 excess, namely (4,1') and (2,1',1,1') where odd parts on even positions are marked by a quote.

Cf. A000720, A077028.

Table[ CoefficientList[ x^Floor[(3+Sqrt[1+8*n])/4]* Tr[x^Tr[(-1)^Mod[Flatten[Position[#,_?OddQ]],2]]&/@Partitions[n]],x],{n,0,12}]; (* or *)
a712[n_Integer]:= a712[n] =If[n<0, 0, (# . Reverse[#])& [PartitionsP[ Range[0, n] ]]]; Table[If[Mod[n+k,2]==1,0,a712[-1+Max[0,(2+n-k*(2*k+1))/2]]],{n,0,12},{k,-Floor[(3+Sqrt[1+8*n])/4],Floor[(-1+Sqrt[1+8*n])/4]}]

A329854 Triangle read by rows: T(n,k) = ((n - k)*(n + k - 1) + 2)/2, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 7, 6, 4, 1, 11, 11, 10, 8, 5, 1, 16, 16, 15, 13, 10, 6, 1, 22, 22, 21, 19, 16, 12, 7, 1, 29, 29, 28, 26, 23, 19, 14, 8, 1, 37, 37, 36, 34, 31, 27, 22, 16, 9, 1, 46, 46, 45, 43, 40, 36, 31, 25, 18, 10, 1, 56, 56, 55, 53, 50, 46, 41, 35, 28, 20, 11, 1

Offset: 0

Werner Schulte, Nov 22 2019

This triangle equals A309559 with reversed rows and supplemented main diagonal (all terms are 1).
There are two lower triangular matrices M and N so that the matrix product M * N equals T (seen as a matrix).
           / 1             \                 / 1             \
           | 0 1           |                 | 1 1           |
           | 0 1 1         |                 | 1 1 1         |
  M(n,k) = | 0 1 2 1       |        N(n,k) = | 1 1 1 1       |
           | 0 1 2 3 1     |                 | 1 1 1 1 1     |
           | 0 1 2 3 4 1   |                 | 1 1 1 1 1 1   |
           \ . . . . . . . /                 \ . . . . . . . /
  The matrix product N * M equals the rascal triangle A077028 (seen as a matrix).

			The triangle T(n,k) starts:
n \ k :   0    1    2    3    4    5    6    7    8    9   10   11
==================================================================
   0  :   1
   1  :   1    1
   2  :   2    2    1
   3  :   4    4    3    1
   4  :   7    7    6    4    1
   5  :  11   11   10    8    5    1
   6  :  16   16   15   13   10    6    1
   7  :  22   22   21   19   16   12    7    1
   8  :  29   29   28   26   23   19   14    8    1
   9  :  37   37   36   34   31   27   22   16    9    1
  10  :  46   46   45   43   40   36   31   25   18   10    1
  11  :  56   56   55   53   50   46   41   35   28   20   11    1
etc.

Row sums equal A116731(n+1).
Row sums apart from column 0 equal A081489.
Cf. A077028, A309559.

O.g.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = ((t^2+(1-t)^2) * (1-x*t) + x * t^2 * (1-t)) / ((1-t)^3 * (1-x*t)^2).
G.f. of column k: Sum_{n>=k} T(n,k) * t^n = t^k * (t^2/(1-t)^3 + 1/(1-t) + k*t/(1-t)^2) for k >= 0.
T(n,k) = 1 + T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 0 < k < n with initial values T(n,0) = (n*(n-1)+2)/2 and T(n,n) = 1 for n >= 0.
T(n,k) = (2 + T(n-1,k-1) * T(n-1,k+1)) / T(n-2,k) for 0 < k < n-1 with initial values given above and T(n,n-1) = n for n > 0.
Referring to the triangle M(n,k) (see comments), we get:
  (1) Sum_{k=0..n} (k+1) * M(n,k) = A116731(n+1) for n >= 0;
  (2) Sum_{k=1..n} k * M(n,k) = A081489(n) for n >= 1.
T(n,k) = T(n-1,k-1) + n-k for 0 < k <= n with initial values T(n,0) = (n*(n-1)+2)/2 for n >= 0.
T(n,k) = 2 * T(n-1,k-1) - T(n-2,k-2) for 1 < k <= n with initial values T(0,0) = 1 and T(n,0) = T(n,1) = (n*(n-1)+2)/2 for n > 0.

cp's OEIS Frontend

A224937 Number of partitions of n having T(n,k) odd parts in excess on even places over odd places.

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