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A259485 Number of n X n connected Tesler matrices.

%I A259485 #18 Nov 14 2020 07:03:06
%S A259485 1,1,4,27,275,4066,85888,2567269,107630237,6269269823,502429080919,
%T A259485 54869692738326,8091237358339821,1597342350434681954,
%U A259485 418809228874760212806,144760685900877097431589,65510311668753649557469187,38566383210089506976493649269,29359678772700284486457832056879
%N A259485 Number of n X n connected Tesler matrices.
%C A259485 Number of n X n upper triangular matrices A of nonnegative integers such that a_1i + a_2i + ... + a_{i-1,i} - a_ii - a_{i,i+1} - ... - a_in = -1, with lowest lattice path above the positive entries not touching the diagonal.
%H A259485 D. Armstrong, A. Garsia, J. Haglund, B. Rhoades and B. Sagan, <a href="https://www.math.upenn.edu/~jhaglund/preprints/TeslerMatrices.pdf">Combinatorics of Tesler matrices in the theory of parking functions and diagonal harmonics</a>, J. of Combin., 3(3):451-494, 2012.
%H A259485 D. Armstrong, <a href="http://www.math.miami.edu/~armstrong/Talks/Tesler_Saganfest.pdf">Tesler Matrices</a>, slides, Saganfest, March 2014.
%F A259485 a(n) = A008608(n) - Sum_{i=1..n-1} A008608(n-i)*a(i).
%e A259485 For n = 3 the a(3) = 4 matrices are [[0,1,0],[0,1,1],[0,0,2]], [[0,1,0],[0,0,2],[0,0,3]], [[0,0,1],[0,1,0],[0,0,2]], [[0,0,1],[0,0,1],[0,0,3]].
%p A259485 multcoeff:=proc(n, f, coeffv, k)
%p A259485    local i, currcoeff;
%p A259485    currcoeff:=f;
%p A259485    for i from 1 to n do
%p A259485       currcoeff:=`if`(coeffv[i]=0, coeff(series(currcoeff, x[i], k), x[i], 0), coeff(series(currcoeff, x[i], k), x[i]^coeffv[i]));
%p A259485    end do;
%p A259485    return currcoeff;
%p A259485 end proc:
%p A259485 F:=n->mul(mul((1-x[i]*x[j]^(-1))^(-1), j=i+1..n), i=1..n):
%p A259485 b := n -> multcoeff(n+1, F(n+1), [seq(1, i=1..n), -n], n+2):
%p A259485 a := n -> `if`(n=1,1,b(n)-add(b(n-i)*a(i),i=1..n-1)):
%p A259485 seq(a(i), i=2..6)
%t A259485 b[n_, i_, l_] := b[n, i, l] = With[{m = Length[l]}, If[m == 0, 1, If[i == 0, b[l[[1]] + 1, m - 1, ReplacePart[l, 1 -> Sequence[]]], Sum[b[n - j, i - 1, ReplacePart[l, i -> l[[i]] + j]], {j, 0, n}]]]];
%t A259485 c[n_] := b[1, n - 1, Array[0&, n - 1]];
%t A259485 a[n_] := a[n] = c[n] - Sum[c[n - i] a[i], {i, 1, n - 1}];
%t A259485 Table[Print[n, " ", a[n]]; a[n], {n, 1, 19}] (* _Jean-François Alcover_, Nov 13 2020, after _Alois P. Heinz_ in A008608 *)
%Y A259485 Cf. A008608, A259666.
%K A259485 nonn
%O A259485 1,3
%A A259485 _Alejandro H. Morales_, Jun 28 2015
%E A259485 a(15)-a(19) from _Alois P. Heinz_, Jul 05 2015