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.)
FORMULA
T(n,k) is the coefficient of x^n in 1/((1-kx)(1-(k-1)x)^2...(1-x)^2).
EXAMPLE
T(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.
MATHEMATICA
f[k_]:=Product[1-j x, {j, k}]; T[n_, k_]:=Coefficient[Series[1/f[k]/f[k-1], {x, 0, n}], x, n]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Don Knuth, May 26 2017
STATUS
approved