A389686 Array read by antidiagonals: T(n,k) is the number of nondecreasing integer sequences of length k with sum zero and sum of absolute values 2n.

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 3, 1, 0, 0, 1, 1, 4, 3, 1, 0, 0, 1, 1, 4, 6, 5, 1, 0, 0, 1, 1, 4, 8, 11, 5, 1, 0, 0, 1, 1, 4, 9, 17, 13, 7, 1, 0, 0, 1, 1, 4, 9, 22, 23, 22, 7, 1, 0, 0, 1, 1, 4, 9, 24, 33, 44, 24, 9, 1, 0, 0, 1, 1, 4, 9, 25, 41, 67, 54, 35, 9, 1, 0, 0

Offset

0,18

Links

Andrew Howroyd, Table of n, a(n) for n = 0..5150 (first 101 antidiagonals)

Formula

G.f.: HadamardSquare(Product_{k=1..n} 1/(1 - y*x^k))/(1 - y), where HadamardSquare(g) is the termwise product of the series g with itself.

Example

Array begins:
================================================
n\k  | 0 1 2  3  4   5   6   7   8    9   10 ...
-----+------------------------------------------
   0 | 1 1 1  1  1   1   1   1   1    1    1 ...
   1 | 0 0 1  1  1   1   1   1   1    1    1 ...
   2 | 0 0 1  3  4   4   4   4   4    4    4 ...
   3 | 0 0 1  3  6   8   9   9   9    9    9 ...
   4 | 0 0 1  5 11  17  22  24  25   25   25 ...
   5 | 0 0 1  5 13  23  33  41  46   48   49 ...
   6 | 0 0 1  7 22  44  67  87 103  113  118 ...
   7 | 0 0 1  7 24  54  92 130 163  189  207 ...
   8 | 0 0 1  9 35  85 156 234 307  367  414 ...
   9 | 0 0 1  9 39 107 214 344 478  598  697 ...
  10 | 0 0 1 11 52 150 318 542 791 1031 1240 ...
   ...
The T(3,4) = 6 sequences are (-3,1,1,1), (-3,0,1,2), (-3,0,0,3), (-2,-1,1,2), (-2,-1,0,3), (-1,-1,-1,3).

Prog

(PARI)
Mtx(n, m=n)=my(y='y+O('y^(1+m)), g=1/prod(k=1, n, 1-y*x^k, 1 + O(x*x^n)));  Mat([Vec(p+O('y^(1+m)), -m-1) | p<-Col(serconvol(g, g)/(1-y))]~)
{ my(A=Mtx(10, 10)); for(i=1, #A~, print(A[i, ])) }

Crossrefs

Columns 0..50 are A000007, A000007, A000012, A109613, A158138, A158139, A158140, A158141, A158142, A158143-A158184.

Sequence in context: A205531 A269246 A334566 * A342270 A330248 A247505

Adjacent sequences: A389683 A389684 A389685 * A389687 A389688 A389689

Keyword

nonn,tabl

Author

Andrew Howroyd, Oct 10 2025

Status

approved