OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
EXAMPLE
Triangle begins:
1;
2, 1;
4, 8, 1;
8, 32, 18, 1;
16, 124, 140, 32, 1;
32, 444, 888, 432, 50, 1;
64, 1568, 5016, 4196, 1060, 72, 1;
128, 5440, 26796, 34732, 15064, 2224, 98, 1;
...
The T(3,1) = 4 multiset partitions are {{1,1,1}}, {{1,1,2}}, {{1,2,2}}, {{1,2,3}}.
The T(3,2) = 8 multiset partitions are {{1},{1,1}}, {{1},{2,2}}, {{2},{1,2}}, {{1},{1,2}}, {{2},{1,1}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}.
The T(3,3) = 1 multiset partition is {{1},{2},{3}}.
MATHEMATICA
B[n_, k_] := Sum[Binomial[r, k] (-1)^(r-k), {r, k, n}];
row[n_] := Sum[B[n, j] SeriesCoefficient[ Product[(1 + x^k y)^Binomial[k + j - 1, j - 1], {k, 1, n}], {x, 0, n}], {j, 1, n}]/y + O[y]^n // CoefficientList[#, y]&;
Array[row, 10] // Flatten (* Jean-François Alcover, Dec 17 2020, after Andrew Howroyd *)
PROG
(PARI) \\ here B(n, k) is A239473(n, k)
B(n, k)={sum(r=k, n, binomial(r, k)*(-1)^(r-k))}
Row(n)={Vecrev(sum(j=1, n, B(n, j)*polcoef(prod(k=1, n, (1 + x^k*y + O(x*x^n))^binomial(k+j-1, j-1)), n))/y)}
{ for(n=1, 10, print(Row(n))) }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Dec 31 2019
STATUS
approved