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A321964
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Array of sequences read by descending antidiagonals, row A(n) is Stieltjes generated from the sequence n, n+1, n+2, n+3, ....
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4
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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
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OFFSET
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0,8
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LINKS
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FORMULA
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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.
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EXAMPLE
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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;
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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