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A321964 Array of sequences read by descending antidiagonals, row A(n) is Stieltjes generated from the sequence n, n+1, n+2, n+3, .... 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Table of n, a(n) for n=0..54.

P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), pp. 125-161.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009.

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.

EXAMPLE

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;

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-Fran├žois Alcover, Jan 07 2019, translated from Maple *)

PROG

(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))

CROSSREFS

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

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

Cf. A321960.

Sequence in context: A344499 A284799 A111106 * A197819 A232006 A202820

Adjacent sequences:  A321961 A321962 A321963 * A321965 A321966 A321967

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Dec 26 2018

STATUS

approved

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Last modified October 25 11:29 EDT 2021. Contains 348251 sequences. (Running on oeis4.)