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A259786 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.

%I A259786 #21 Mar 03 2024 17:29:16
%S A259786 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 A259786 260,526,771,878,811,627,416,238,118,50,18,5,1,1,21,175,805,2436,5362,
%U A259786 9123,12568,14465,14289,12345,9483,6534,4071,2297,1176,542,224,81,25,6,1
%N A259786 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 A259786 For the definition of Tesler matrices see A008608.
%H A259786 Alois P. Heinz, <a href="/A259786/b259786.txt">Rows n = 1..17, flattened</a>
%F A259786 Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A259787(n).
%e A259786 Triangle T(n,k) begins:
%e A259786   1;
%e A259786   1,  1;
%e A259786   1,  3,  2,   1;
%e A259786   1,  6, 11,  11,   7,   3,   1;
%e A259786   1, 10, 35,  65,  81,  71,  50,  27,  12,   4,   1;
%e A259786   1, 15, 85, 260, 526, 771, 878, 811, 627, 416, 238, 118, 50, 18, 5, 1;
%e A259786   ...
%p A259786 b:= proc(n, i, l) option remember; (m-> `if`(m=0, 1, expand(
%p A259786       `if`(i=0, x^(l[1]+1)*b(l[1]+1, m-1, subsop(1=NULL, l)), add(
%p A259786        b(n-j, i-1, subsop(i=l[i]+j, l)), j=0..n)))))(nops(l))
%p A259786     end:
%p A259786 T:= n->(p->seq(coeff(p, x, i), i=n-1..degree(p)))(b(1, n-1, [0$(n-1)])):
%p A259786 seq(T(n), n=1..8);
%t A259786 b[n_, i_, l_] := b[n, i, l] = Function[m, If[m == 0, 1, Expand[
%t A259786      If[i == 0, x^(l[[1]] + 1)*b[l[[1]] + 1, m - 1,
%t A259786      ReplacePart[l, 1 -> Nothing]], Sum[b[n - j, i - 1,
%t A259786      ReplacePart[l, i -> l[[i]] + j]], {j, 0, n}]]]]][Length[l]];
%t A259786 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, n - 1,
%t A259786      Exponent[p, x]}]][b[1, n - 1, Table[0, {n - 1}]]];
%t A259786 Table[T[n], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)
%Y A259786 Row sums give A008608.
%Y A259786 Cf. A000217, A259787.
%K A259786 nonn,tabf
%O A259786 1,5
%A A259786 _Alois P. Heinz_, Jul 05 2015