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A263756 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n with sum of descent bottoms equal to k.

%I A263756 #26 May 26 2023 05:21:34
%S A263756 1,1,1,1,1,3,1,1,1,7,3,8,3,1,1,1,15,7,34,18,14,18,8,3,1,1,1,31,15,122,
%T A263756 72,69,147,83,71,33,45,18,8,3,1,1,1,63,31,406,252,263,822,544,554,399,
%U A263756 613,351,307,160,102,96,45,18,8,3,1,1,1,127,63,1298,828
%N A263756 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n with sum of descent bottoms equal to k.
%C A263756 Row sums give A000142.
%H A263756 Alois P. Heinz, <a href="/A263756/b263756.txt">Rows n = 0..23, flattened</a>
%H A263756 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000154">The sum of the descent bottoms of a permutations</a>.
%e A263756 Triangle begins:
%e A263756   1;
%e A263756   1;
%e A263756   1,1;
%e A263756   1,3,1,1;
%e A263756   1,7,3,8,3,1,1;
%e A263756   1,15,7,34,18,14,18,8,3,1,1;
%e A263756   1,31,15,122,72,69,147,83,71,33,45,18,8,3,1,1;
%e A263756   ...
%p A263756 b:= proc(s) option remember; (n-> `if`(n=0, 1, expand(
%p A263756       add(b(s minus {j})*`if`(j<n, x^j, 1), j=s))))(nops(s))
%p A263756     end:
%p A263756 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
%p A263756 seq(T(n), n=0..9);  # _Alois P. Heinz_, Oct 25 2015, revised, Jan 31 2023
%t A263756 b[s_] := b[s] = With[{n = Length[s]}, If[n == 0, 1, Expand[       Sum[b[s~Complement~{j}]*If[j < n, x^j, 1], {j, s}]]]];
%t A263756 T[n_] := CoefficientList[b[Range[n]], x];
%t A263756 Table[T[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, May 26 2023, after _Alois P. Heinz_ *)
%Y A263756 Cf. A000142, A263753.
%K A263756 nonn,tabf
%O A263756 0,6
%A A263756 _Christian Stump_, Oct 19 2015
%E A263756 Two terms (for rows 0 and 1) prepended and more terms from _Alois P. Heinz_, Oct 25 2015