A287532 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals upwards, where A(n,k) = sum of unimodal products of length n and bound k.

1, 1, 1, 1, 4, 1, 1, 11, 9, 1, 1, 26, 50, 16, 1, 1, 57, 222, 150, 25, 1, 1, 120, 867, 1080, 355, 36, 1, 1, 247, 3123, 6627, 3775, 721, 49, 1, 1, 502, 10660, 36552, 33502, 10626, 1316, 64, 1, 1, 1013, 35064, 187000, 262570, 128758, 25676, 2220, 81, 1

Offset

0,5

Comments

A unimodal product of length n and parameter k is a product of positive integers a_1 ... a_m ... a_n where a_1 <= ... <= a_m <= k and k >= a_m >= ... >= a_n; furthermore we consider each choice of m to give a distinct product, unless a_m=k. (See the example.)

Links

Table of n, a(n) for n=0..54.

Formula

A(n,k) is the coefficient of x^n in 1/((1-k*x) * (1-(k-1)*x)^2 * ... * (1-x)^2).

A(n,k) = Sum_{j=0..n} Stirling2(j+k-1,k-1) * Stirling2(n-j+k,k) for k >= 1. - Seiichi Manyama, May 14 2025

Example

A(2,3)=50 because of the products 1*1,1*1,1*1 [m=0,1,2]; 1*2,1*2 [m=1,2]; 1*3; 2*1,2*1 [m=0,1]; 2*2,2*2,2*2 [m=0,1,2]; 2*3; 3*1; 3*2; 3*3; total 50.
Square array begins:
  n\k| 1,   2,    3,     4,      5,       6, ...
  ---+------------------------------------------
   0 | 1,   1,    1,     1,      1,       1, ...
   1 | 1,   4,    9,    16,     25,      36, ...
   2 | 1,  11,   50,   150,    355,     721, ...
   3 | 1,  26,  222,  1080,   3775,   10626, ...
   4 | 1,  57,  867,  6627,  33502,  128758, ...
   5 | 1, 120, 3123, 36552, 262570, 1360128, ...
  ...

Mathematica

f[k_]:=Product[1-j x, {j, k}]; A[n_, k_]:=Coefficient[Series[1/f[k]/f[k-1], {x, 0, n}], x, n]

Prog

(PARI) a(n, k) = sum(j=0, n, stirling(j+k-1, k-1, 2)*stirling(n-j+k, k, 2)); \\ Seiichi Manyama, May 14 2025

Crossrefs

A(n,n) gives A383883.

Columns k=5..6 give A383892, A383893.

Cf. A000290, A000295, A222993, A223069.

Sequence in context: A147564 A090981 A087903 * A112500 A152938 A154096

Adjacent sequences: A287529 A287530 A287531 * A287533 A287534 A287535

Keyword

nonn,tabl

Author

Don Knuth, May 26 2017

Status

approved