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A134431 Triangle read by rows: T(n,k) is the number of arrangements of the set {1,2,...,n} in which the sum of the entries is equal to k (n >= 0, k >= 0; to n=0 there corresponds the empty set).

%I A134431 #12 Feb 19 2020 07:21:44
%S A134431 1,1,1,1,1,1,2,1,1,1,3,2,2,6,1,1,1,3,3,4,8,8,6,6,24,1,1,1,3,3,5,10,10,
%T A134431 14,14,36,30,30,24,24,120,1,1,1,3,3,5,11,12,16,22,44,44,66,60,78,174,
%U A134431 168,144,144,120,120,720,1,1,1,3,3,5,11,13,18,24,52,52,80,98,120,234
%N A134431 Triangle read by rows: T(n,k) is the number of arrangements of the set {1,2,...,n} in which the sum of the entries is equal to k (n >= 0, k >= 0; to n=0 there corresponds the empty set).
%C A134431 Row n has 1 + n(n+1)/2 terms (n >= 0). Row sums yield the arrangement numbers (A000522). T(n, n(n+1)/2) = n!. Sum_{k=0..n(n+1)/2} k*T(n,k) = A134432(n).
%H A134431 Alois P. Heinz, <a href="/A134431/b134431.txt">Rows n = 0..48, flattened</a>
%F A134431 The row generating polynomials P[n](t) are equal to Q[n](t,1), where the polynomials Q[n](t,x) are defined by Q[0]=1 and Q[n]=Q[n-1] + xt^n (d/dx)xQ[n-1]. [Q[n](t,x) is the bivariate generating polynomial of the arrangements of {1,2,...,n}, where t (x) marks the sum (number) of the entries; for example, Q[2](t,x)=1+tx + t^2*x + 2t^3*x^2, corresponding to: empty, 1, 2, 12 and 21, respectively.]
%e A134431 T(4,7)=8 because we have 34,43 and the six permutations of {1,2,4}.
%e A134431 Triangle starts:
%e A134431   1;
%e A134431   1, 1;
%e A134431   1, 1, 1, 2;
%e A134431   1, 1, 1, 3, 2, 2, 6;
%e A134431   1, 1, 1, 3, 3, 4, 8, 8, 6, 6, 24;
%p A134431 Q[0]:=1: for n to 7 do Q[n]:=sort(simplify(Q[n-1]+t^n*x*(diff(x*Q[n-1],x))), t) end do: for n from 0 to 7 do P[n]:=sort(subs(x=1,Q[n])) end do: for n from 0 to 7 do seq(coeff(P[n],t,j),j=0..(1/2)*n*(n+1)) end do; # yields sequence in triangular form
%p A134431 # second Maple program:
%p A134431 b:= proc(n, s, t) option remember;
%p A134431       `if`(n=0, t!*x^s, b(n-1, s, t)+b(n-1, s+n, t+1))
%p A134431     end:
%p A134431 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
%p A134431 seq(T(n), n=0..8);  # _Alois P. Heinz_, Dec 22 2017
%t A134431 b[n_, s_, t_] := b[n, s, t] = If[n == 0, t!*x^s, b[n - 1, s, t] + b[n - 1, s + n, t + 1]];
%t A134431 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]] @ b[n, 0, 0];
%t A134431 T /@ Range[0, 8] // Flatten (* _Jean-François Alcover_, Feb 19 2020, after _Alois P. Heinz_ *)
%Y A134431 Cf. A000522, A134432.
%K A134431 nonn,tabf
%O A134431 0,7
%A A134431 _Emeric Deutsch_, Nov 16 2007