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Number T(n,k) of n X n Tesler matrices of nonnegative integers with element sum n+k; triangle T(n,k), n>=1, 0<=k<=n*(n-1)/2, read by rows.
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%I #21 Mar 03 2024 17:29:16

%S 1,1,1,1,3,2,1,1,6,11,11,7,3,1,1,10,35,65,81,71,50,27,12,4,1,1,15,85,

%T 260,526,771,878,811,627,416,238,118,50,18,5,1,1,21,175,805,2436,5362,

%U 9123,12568,14465,14289,12345,9483,6534,4071,2297,1176,542,224,81,25,6,1

%N Number T(n,k) of n X n Tesler matrices of nonnegative integers with element sum n+k; triangle T(n,k), n>=1, 0<=k<=n*(n-1)/2, read by rows.

%C For the definition of Tesler matrices see A008608.

%H Alois P. Heinz, <a href="/A259786/b259786.txt">Rows n = 1..17, flattened</a>

%F Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A259787(n).

%e Triangle T(n,k) begins:

%e 1;

%e 1, 1;

%e 1, 3, 2, 1;

%e 1, 6, 11, 11, 7, 3, 1;

%e 1, 10, 35, 65, 81, 71, 50, 27, 12, 4, 1;

%e 1, 15, 85, 260, 526, 771, 878, 811, 627, 416, 238, 118, 50, 18, 5, 1;

%e ...

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

%p `if`(i=0, x^(l[1]+1)*b(l[1]+1, m-1, subsop(1=NULL, l)), add(

%p b(n-j, i-1, subsop(i=l[i]+j, l)), j=0..n)))))(nops(l))

%p end:

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

%p seq(T(n), n=1..8);

%t b[n_, i_, l_] := b[n, i, l] = Function[m, If[m == 0, 1, Expand[

%t If[i == 0, x^(l[[1]] + 1)*b[l[[1]] + 1, m - 1,

%t ReplacePart[l, 1 -> Nothing]], Sum[b[n - j, i - 1,

%t ReplacePart[l, i -> l[[i]] + j]], {j, 0, n}]]]]][Length[l]];

%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, n - 1,

%t Exponent[p, x]}]][b[1, n - 1, Table[0, {n - 1}]]];

%t Table[T[n], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)

%Y Row sums give A008608.

%Y Cf. A000217, A259787.

%K nonn,tabf

%O 1,5

%A _Alois P. Heinz_, Jul 05 2015