A259841 - OEIS

%I #17 Mar 03 2024 17:29:03

%S 1,3,1,15,5,2,117,37,17,7,1367,418,189,100,40,23329,7027,3058,1688,

%T 939,357,570933,171428,72194,39274,24050,13429,4820,19740068,5948380,

%U 2449366,1293768,807576,517548,283510,96030

%N 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 For the definition of Tesler matrices see A008608.

%C Sum_{k=1..n} k * T(n,k) = A259787(n).

%H Alois P. Heinz, <a href="/A259841/b259841.txt">Rows n = 1..20, flattened</a>

%e 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 Triangle T(n,k) begins:

%e        1;

%e        3,      1;

%e       15,      5,     2;

%e      117,     37,    17,     7;

%e     1367,    418,   189,   100,    40;

%e    23329,   7027,  3058,  1688,   939,   357;

%e   570933, 171428, 72194, 39274, 24050, 13429, 4820;

%e   ...

%p g:= u-> `if`(u=0, 0, x^u):

%p b:= proc(n, i, l) option remember; (m->`if`(m=0, [1, g(n)], `if`(i=0,

%p      (p->p+[0, p[1]*g(n)])(b(l[1]+1, m-1, subsop(1=NULL, l))), add(

%p      (p->p+[0, p[1]*g(j)])(b(n-j, i-1, subsop(i=l[i]+j, l)))

%p       , j=0..n))))(nops(l))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(1,n-1,[0$(n-1)])[2]):

%p seq(T(n), n=1..10);

%t g[u_] := If[u == 0, 0, x^u];

%t 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 T[n_] := Table[Coefficient[#, x, i], {i, 1, n}] & [b[1, n-1, Table[0, {n-1}]][[2]]];

%t Array[T, 10] // Flatten (* _Jean-François Alcover_, Oct 28 2020, after Maple *)

%Y Main diagonal gives A008608(n-1) for n>1.

%Y Column k=1 gives A259843.

%Y Row sums give A259842.

%Y Cf. A259787.

%K nonn,tabl

%O 1,2

%A _Alois P. Heinz_, Jul 06 2015