This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
First few rows of the array start: [0] 1, 0, 0, 0, 0, 0, 0, 0, ... A000007 [1] 1, 1, 3, 15, 105, 945, 10395, 135135, ... A001147 [2] 1, 2, 10, 74, 706, 8162, 110410, 1708394, ... A000698 [3] 1, 3, 21, 207, 2529, 36243, 591381, 10786527, ... A167872 [4] 1, 4, 36, 444, 6636, 114084, 2194596, 46460124, ... A321963 [5] 1, 5, 55, 815, 14425, 289925, 6444175, 155928575, ... [6] 1, 6, 78, 1350, 27630, 636390, 16074990, 438572070, ... Seen as triangle: [0] 1; [1] 0, 1; [2] 0, 1, 1; [3] 0, 3, 2, 1; [4] 0, 15, 10, 3, 1; [5] 0, 105, 74, 21, 4, 1; [6] 0, 945, 706, 207, 36, 5, 1; [7] 0, 10395, 8162, 2529, 444, 55, 6, 1; [8] 0, 135135, 110410, 36243, 6636, 815, 78, 7, 1;
JacobiCF := proc(a, b, p:=2) local m, k;
m := 1;
for k from nops(a) by -1 to 1 do
m := 1 - b[k]*x - a[k]*x^p/m od;
return 1/m end:
JacobiGF := proc(a, b, p:=2) local cf, l, ser;
cf := JacobiCF(a, b, p);
l := min(nops(a), nops(b));
ser := series(cf, x, l);
seq(coeff(ser, x, n), n = 0..l-1) end:
JacobiSquare := proc(a, p:=2) local cf, ser;
cf := JacobiCF(a, a, p);
ser := series(cf, x, nops(a));
seq(coeff(ser, x, n), n = 0..nops(a)-1) end:
StieltjesGF := proc(a, p:=2) local z, cf, ser;
z := [seq(0, n = 1..nops(a))];
cf := JacobiCF(a, z, p);
ser := series(cf, x, nops(a));
seq(coeff(ser, x, n), n = 0..nops(a)-1) end:
s := n -> [seq(n+k, k = 0..9)]:
Trow := n -> StieltjesGF(s(n), 1):
for n from 0 to 6 do lprint(Trow(n)) od;
nmax = 9;
JacobiCF[a_, b_, p_:2] := Module[{m, k}, m = 1; For[k = Length[a] , k >= 1, k--, m = 1 - b[[k]]*x - a[[k]]*x^p/m ]; 1/m];
JacobiGF[a_, b_, p_:2] := Module[{cf, l, ser}, cf = JacobiCF[a, b, p]; l = Min[Length[a], Length[b]]; ser = Series[cf, {x, 0, l}]; CoefficientList[ ser, x]];
JacobiSquare[a_, p_:2] := Module[{cf, ser}, cf = JacobiCF[a, a, p]; ser = Series[cf, {x, 0, Length[a]}]; CoefficientList[ser, x]];
StieltjesGF[a_, p_:2] := Module[{z, cf, ser}, z = Table[0, Length[a]]; cf = JacobiCF[a, z, p]; ser = Series[cf, {x, 0, Length[a]}]; CoefficientList[ ser, x]];
s[n_] := Table[n + k, {k, 0, nmax}];
Trow[0] = Table[Boole[k == 0], {k, 0, nmax}];
Trow[n_] := Trow[n] = StieltjesGF[s[n], 1] ;
T[n_, k_] := Trow[n][[k + 1]];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 07 2019, translated from Maple *)
# uses[StieltjesGF from A321960] def Trow(n, dim): return StieltjesGF(lambda k: n+k, dim, p=1) for n in (0..7): print(Trow(n, 9))