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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.

%I A256193 #40 Feb 16 2025 08:33:25
%S A256193 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,
%T A256193 54,89,91,63,29,8,1,22,86,158,186,155,92,37,9,1,30,128,262,351,342,
%U A256193 247,129,46,10,1,42,192,428,635,700,590,376,175,56,11,1
%N 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.
%H A256193 Alois P. Heinz, <a href="/A256193/b256193.txt">Rows n = 0..140, flattened</a>
%H A256193 William Dugan, Sam Glennon, Paul E. Gunnells, and Einar Steingrimsson, <a href="https://arxiv.org/abs/1702.02446">Tiered trees, weights, and q-Eulerian numbers</a>, arXiv:1702.02446 [math.CO], Feb 2017.
%H A256193 Emmy Huang and Ray Tang, <a href="https://arxiv.org/abs/2310.14385">Minimum Decomposition on Maxmin Trees</a>, arXiv:2310.14385 [math.CO], 2023.
%H A256193 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%H A256193 Wikipedia, <a href="https://en.wikipedia.org/wiki/Q-Pochhammer_symbol">q-Pochhammer symbol</a>
%F A256193 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
%F A256193 Sum_{k=0..n} k * T(n,k) = A278464(n). - _Alois P. Heinz_, Nov 22 2016
%e A256193 T(3,0) = 3: 111, 21, 3.
%e A256193 T(3,1) = 6: 1'11, 11'1, 111', 2'1, 21', 3'.
%e A256193 T(3,2) = 4: 1'1'1, 1'11', 11'1', 2'1'.
%e A256193 T(3,3) = 1: 1'1'1'.
%e A256193 Triangle T(n,k) begins:
%e A256193    1;
%e A256193    1,   1;
%e A256193    2,   3,   1;
%e A256193    3,   6,   4,   1;
%e A256193    5,  12,  11,   5,   1;
%e A256193    7,  20,  24,  16,   6,   1;
%e A256193   11,  35,  49,  41,  22,   7,   1;
%e A256193   15,  54,  89,  91,  63,  29,   8,   1;
%e A256193   22,  86, 158, 186, 155,  92,  37,   9,  1;
%e A256193   30, 128, 262, 351, 342, 247, 129,  46, 10,  1;
%e A256193   42, 192, 428, 635, 700, 590, 376, 175, 56, 11,  1;
%e A256193   ...
%p A256193 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A256193       `if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
%p A256193        binomial(j, t), t=0..j), j=0..n/i))))
%p A256193     end:
%p A256193 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A256193 seq(T(n), n=0..12);
%t A256193 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_ *)
%t A256193 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 *)
%Y A256193 Column k=0-10 gives: A000041, A006128, A258472, A258473, A258474, A258475, A258476, A258477, A258478, A258479, A258480.
%Y A256193 T(2n,n) gives A258471.
%Y A256193 Row sums give A070933.
%Y A256193 Cf. A278464.
%K A256193 nonn,tabl
%O A256193 0,4
%A A256193 _Alois P. Heinz_, Mar 19 2015