A259666 Number of n X n prime Tesler matrices.

1, 1, 3, 18, 181, 2788, 62590, 1989540, 87979661, 5349559222, 443306080232, 49679250634068, 7473835936432840, 1498682325685621140, 397803907069442925517, 138847938093177059278212, 63325340852730727078521540, 37513306417359729218973719474, 28701720575221087513434901774347

Offset

1,3

Comments

Number of n X n upper triangular matrices A of nonnegative integers such that a_{1,i} + a_{2,i} + ... + a_{i-1,i} - a_{i,i} - a_{i,i+1} - ... - a_{i,n} = -1, whose simple graph G with vertices 1,2,3..,n and edges (i,j) if a_{i,j} > 0 is connected.

Links

Table of n, a(n) for n=1..19.

A. Garsia and J. Haglund, A polynomial expression for the character of diagonal harmonics, Ann. Comb., to appear, 2015.

Formula

E.g.f.: 1 + log( 1+ sum(n>=1, A008608(n) * x^n / n! ) ).

Example

Example: For n =3 the a(3) = 3 matrices are [[0,1,0],[0,1,1],[0,0,2]], [[0,1,0],[0,0,2],[0,0,3]], [[0,0,1],[0,0,1],[0,0,3]].
E.g.f.: 1 + x+(1/2)*x^2+(3/6)*x^3+(18/24)*x^4+(181/120)*x^5+(2788/720)*x^6 + ...

Maple

multcoeff:=proc(n, f, coeffv, k)
   local i, currcoeff;
   currcoeff:=f;
   for i from 1 to n do
      currcoeff:=`if`(coeffv[i]=0, coeff(series(currcoeff, x[i], k), x[i], 0), coeff(series(currcoeff, x[i], k), x[i]^coeffv[i]));
   end do;
   return currcoeff;
end proc:
F:=n->mul(mul((1-x[i]*x[j]^(-1))^(-1), j=i+1..n), i=1..n):
b := n -> multcoeff(n+1, F(n+1), [seq(1, i=1..n), -n], n+2):
sa := 1 + log(1+ add(b(n)*x^n/n!, n=1..7)):
a := n -> n!*coeff(series(sa, x, n+1), x, n):
seq(a(i), i=1..6);

Mathematica

b[n_, i_, l_] := b[n, i, l] = Function[{m}, 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}]]]][Length[l]];
c[n_] := b[1, n-1, Array[0&, n-1]];
a[n_] := a[n] = SeriesCoefficient[1 + Log[1 + Sum[c[k] x^k/k!, {k, 1, n}]], {x, 0, n}] n!;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 19}] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz in A008608 *)

Crossrefs

Cf. A008608, A259485.

Sequence in context: A108994 A006472 A132853 * A323502 A365996 A326088

Adjacent sequences: A259663 A259664 A259665 * A259667 A259668 A259669

Keyword

nonn

Author

Alejandro H. Morales, Jul 02 2015

Extensions

a(15)-a(19) from Alois P. Heinz, Jul 05 2015

Status

approved