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%I A259841 #17 Mar 03 2024 17:29:03
%S A259841 1,3,1,15,5,2,117,37,17,7,1367,418,189,100,40,23329,7027,3058,1688,
%T A259841 939,357,570933,171428,72194,39274,24050,13429,4820,19740068,5948380,
%U A259841 2449366,1293768,807576,517548,283510,96030
%N A259841 Number T(n,k) of elements k in all n X n Tesler matrices of nonnegative integers; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%C A259841 For the definition of Tesler matrices see A008608.
%C A259841 Sum_{k=1..n} k * T(n,k) = A259787(n).
%H A259841 Alois P. Heinz, <a href="/A259841/b259841.txt">Rows n = 1..20, flattened</a>
%e A259841 There are two 2 X 2 Tesler matrices: [1,0; 0,1], [0,1; 0,2], containing three 1's and one 2, thus row 2 gives [3, 1].
%e A259841 Triangle T(n,k) begins:
%e A259841        1;
%e A259841        3,      1;
%e A259841       15,      5,     2;
%e A259841      117,     37,    17,     7;
%e A259841     1367,    418,   189,   100,    40;
%e A259841    23329,   7027,  3058,  1688,   939,   357;
%e A259841   570933, 171428, 72194, 39274, 24050, 13429, 4820;
%e A259841   ...
%p A259841 g:= u-> `if`(u=0, 0, x^u):
%p A259841 b:= proc(n, i, l) option remember; (m->`if`(m=0, [1, g(n)], `if`(i=0,
%p A259841      (p->p+[0, p[1]*g(n)])(b(l[1]+1, m-1, subsop(1=NULL, l))), add(
%p A259841      (p->p+[0, p[1]*g(j)])(b(n-j, i-1, subsop(i=l[i]+j, l)))
%p A259841       , j=0..n))))(nops(l))
%p A259841     end:
%p A259841 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(1,n-1,[0$(n-1)])[2]):
%p A259841 seq(T(n), n=1..10);
%t A259841 g[u_] := If[u == 0, 0, x^u];
%t A259841 b[n_, i_, l_] := b[n, i, l] = Function[m, If[m == 0, {1, g[n]}, If[i == 0, # + {0, #[[1]] g[n]} & [b[l[[1]]+1, m-1, ReplacePart[l, 1 -> Nothing]]], Sum[# + {0, #[[1]] g[j]} & [b[n-j, i-1, ReplacePart[l, i -> l[[i]] + j]]], {j, 0, n}]]]][Length[l]];
%t A259841 T[n_] := Table[Coefficient[#, x, i], {i, 1, n}] & [b[1, n-1, Table[0, {n-1}]][[2]]];
%t A259841 Array[T, 10] // Flatten (* _Jean-François Alcover_, Oct 28 2020, after Maple *)
%Y A259841 Main diagonal gives A008608(n-1) for n>1.
%Y A259841 Column k=1 gives A259843.
%Y A259841 Row sums give A259842.
%Y A259841 Cf. A259787.
%K A259841 nonn,tabl
%O A259841 1,2
%A A259841 _Alois P. Heinz_, Jul 06 2015

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