A321964 Array of sequences read by descending antidiagonals, row A(n) is Stieltjes generated from the sequence n, n+1, n+2, n+3, ....

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 10, 3, 1, 0, 105, 74, 21, 4, 1, 0, 945, 706, 207, 36, 5, 1, 0, 10395, 8162, 2529, 444, 55, 6, 1, 0, 135135, 110410, 36243, 6636, 815, 78, 7, 1, 0, 2027025, 1708394, 591381, 114084, 14425, 1350, 105, 8, 1

Offset: 0

Author

Peter Luschny, Dec 26 2018

Keywords

• nonn
• tabl

Examples

			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;
		

Links

• P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), pp. 125-161.
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009.

Crossrefs

Rows of array: A000007, A001147, A000698, A167872, A321963.

Columns include: A014105. Row sums of triangle: A321961.

Cf. A321960.

Programs

Maple

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;

Mathematica

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 *)

Sage

# 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))

Formula

We say a sequence R is Jacobi generated by the sequences U and V if R are the coefficients of the series expansion of the Jacobi continued fraction, recursively defined by m = 1 - V(k)*x - U(k)*x^p/m, starting m = 1 and terminating with 1/m, k iterating downwards from a given length to 1. p is some integer (in the classic case p = 2). R is Stieltjes generated if it is Jacobi generated with V(k) = 0 for all k.

In this array the rows are Stieltjes generated with p = 1 from the sequence s(j) = n + j, j >= 0. T(n, k) = A(n)[k] for n >= 0 and k >= 0.