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A256193
Number T(n,k) of partitions of n into two sorts of parts having exactly k parts of the second sort; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
14
1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 5, 12, 11, 5, 1, 7, 20, 24, 16, 6, 1, 11, 35, 49, 41, 22, 7, 1, 15, 54, 89, 91, 63, 29, 8, 1, 22, 86, 158, 186, 155, 92, 37, 9, 1, 30, 128, 262, 351, 342, 247, 129, 46, 10, 1, 42, 192, 428, 635, 700, 590, 376, 175, 56, 11, 1
OFFSET
0,4
LINKS
William Dugan, Sam Glennon, Paul E. Gunnells, and Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], Feb 2017.
Emmy Huang and Ray Tang, Minimum Decomposition on Maxmin Trees, arXiv:2310.14385 [math.CO], 2023.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol
FORMULA
T(n,k) = [x^k] [q^(n-k)] 1/(q+x; q)_inf = [x^k] [q^(n-k)] 1/(q+x; q)_n, where (x; q)_n is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 22 2016
Sum_{k=0..n} k * T(n,k) = A278464(n). - Alois P. Heinz, Nov 22 2016
EXAMPLE
T(3,0) = 3: 111, 21, 3.
T(3,1) = 6: 1'11, 11'1, 111', 2'1, 21', 3'.
T(3,2) = 4: 1'1'1, 1'11', 11'1', 2'1'.
T(3,3) = 1: 1'1'1'.
Triangle T(n,k) begins:
1;
1, 1;
2, 3, 1;
3, 6, 4, 1;
5, 12, 11, 5, 1;
7, 20, 24, 16, 6, 1;
11, 35, 49, 41, 22, 7, 1;
15, 54, 89, 91, 63, 29, 8, 1;
22, 86, 158, 186, 155, 92, 37, 9, 1;
30, 128, 262, 351, 342, 247, 129, 46, 10, 1;
42, 192, 428, 635, 700, 590, 376, 175, 56, 11, 1;
...
MAPLE
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
binomial(j, t), t=0..j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);
MATHEMATICA
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]* Sum[x^t*Binomial[j, t], {t, 0, j}], {j, 0, n/i}]]]]; T[n_] := Function[ p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
Table[SeriesCoefficient[FunctionExpand[1/QPochhammer[q + x, q, n]], {q, 0, n - k}, {x, 0, k}], {n, 0, 10}, {k, 0, n}] // Column (* Vladimir Reshetnikov, Nov 22 2016 *)
CROSSREFS
T(2n,n) gives A258471.
Row sums give A070933.
Cf. A278464.
Sequence in context: A153861 A118981 A117938 * A101912 A208522 A209569
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 19 2015
STATUS
approved