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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.

%I A287532 #45 May 30 2025 09:41:41
%S A287532 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,
%T A287532 355,36,1,1,247,3123,6627,3775,721,49,1,1,502,10660,36552,33502,10626,
%U A287532 1316,64,1,1,1013,35064,187000,262570,128758,25676,2220,81,1
%N 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.
%C A287532 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.)
%F A287532 A(n,k) is the coefficient of x^n in 1/((1-k*x) * (1-(k-1)*x)^2 * ... * (1-x)^2).
%F A287532 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
%e A287532 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.
%e A287532 Square array begins:
%e A287532   n\k| 1,   2,    3,     4,      5,       6, ...
%e A287532   ---+------------------------------------------
%e A287532    0 | 1,   1,    1,     1,      1,       1, ...
%e A287532    1 | 1,   4,    9,    16,     25,      36, ...
%e A287532    2 | 1,  11,   50,   150,    355,     721, ...
%e A287532    3 | 1,  26,  222,  1080,   3775,   10626, ...
%e A287532    4 | 1,  57,  867,  6627,  33502,  128758, ...
%e A287532    5 | 1, 120, 3123, 36552, 262570, 1360128, ...
%e A287532   ...
%t A287532 f[k_]:=Product[1-j x,{j,k}]; A[n_,k_]:=Coefficient[Series[1/f[k]/f[k-1],{x,0,n}],x,n]
%o A287532 (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
%Y A287532 A(n,n) gives A383883.
%Y A287532 Columns k=5..6 give A383892, A383893.
%Y A287532 Cf. A000290, A000295, A222993, A223069.
%K A287532 nonn,tabl
%O A287532 0,5
%A A287532 _Don Knuth_, May 26 2017