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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, ....

%I A321964 #26 Mar 24 2020 13:25:12
%S A321964 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,
%T A321964 0,10395,8162,2529,444,55,6,1,0,135135,110410,36243,6636,815,78,7,1,0,
%U A321964 2027025,1708394,591381,114084,14425,1350,105,8,1
%N A321964 Array of sequences read by descending antidiagonals, row A(n) is Stieltjes generated from the sequence n, n+1, n+2, n+3, ....
%H A321964 P. Flajolet, <a href="http://dx.doi.org/10.1016/0012-365X(80)90050-3">Combinatorial aspects of continued fractions</a>, Discrete Mathematics 32 (1980), pp. 125-161.
%H A321964 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009.
%F A321964 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.
%F A321964 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.
%e A321964 First few rows of the array start:
%e A321964 [0] 1, 0,  0,    0,     0,      0,        0,         0, ... A000007
%e A321964 [1] 1, 1,  3,   15,   105,    945,    10395,    135135, ... A001147
%e A321964 [2] 1, 2, 10,   74,   706,   8162,   110410,   1708394, ... A000698
%e A321964 [3] 1, 3, 21,  207,  2529,  36243,   591381,  10786527, ... A167872
%e A321964 [4] 1, 4, 36,  444,  6636, 114084,  2194596,  46460124, ... A321963
%e A321964 [5] 1, 5, 55,  815, 14425, 289925,  6444175, 155928575, ...
%e A321964 [6] 1, 6, 78, 1350, 27630, 636390, 16074990, 438572070, ...
%e A321964 Seen as triangle:
%e A321964 [0] 1;
%e A321964 [1] 0,      1;
%e A321964 [2] 0,      1,      1;
%e A321964 [3] 0,      3,      2,     1;
%e A321964 [4] 0,     15,     10,     3,    1;
%e A321964 [5] 0,    105,     74,    21,    4,   1;
%e A321964 [6] 0,    945,    706,   207,   36,   5,  1;
%e A321964 [7] 0,  10395,   8162,  2529,  444,  55,  6, 1;
%e A321964 [8] 0, 135135, 110410, 36243, 6636, 815, 78, 7, 1;
%p A321964 JacobiCF := proc(a, b, p:=2) local m, k;
%p A321964     m := 1;
%p A321964     for k from nops(a) by -1 to 1 do
%p A321964         m := 1 - b[k]*x - a[k]*x^p/m od;
%p A321964     return 1/m end:
%p A321964 JacobiGF := proc(a, b, p:=2) local cf, l, ser;
%p A321964     cf := JacobiCF(a, b, p);
%p A321964     l := min(nops(a), nops(b));
%p A321964     ser := series(cf, x, l);
%p A321964     seq(coeff(ser, x, n), n = 0..l-1) end:
%p A321964 JacobiSquare := proc(a, p:=2) local cf, ser;
%p A321964     cf := JacobiCF(a, a, p);
%p A321964     ser := series(cf, x, nops(a));
%p A321964     seq(coeff(ser, x, n), n = 0..nops(a)-1) end:
%p A321964 StieltjesGF := proc(a, p:=2) local z, cf, ser;
%p A321964     z := [seq(0, n = 1..nops(a))];
%p A321964     cf := JacobiCF(a, z, p);
%p A321964     ser := series(cf, x, nops(a));
%p A321964     seq(coeff(ser, x, n), n = 0..nops(a)-1) end:
%p A321964 s := n -> [seq(n+k, k = 0..9)]:
%p A321964 Trow := n -> StieltjesGF(s(n), 1):
%p A321964 for n from 0 to 6 do lprint(Trow(n)) od;
%t A321964 nmax = 9;
%t A321964 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];
%t A321964 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]];
%t A321964 JacobiSquare[a_, p_:2] := Module[{cf, ser}, cf = JacobiCF[a, a, p]; ser = Series[cf, {x, 0, Length[a]}]; CoefficientList[ser, x]];
%t A321964 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]];
%t A321964 s[n_] := Table[n + k, {k, 0, nmax}];
%t A321964 Trow[0] = Table[Boole[k == 0], {k, 0, nmax}];
%t A321964 Trow[n_] := Trow[n] = StieltjesGF[s[n], 1] ;
%t A321964 T[n_, k_] := Trow[n][[k + 1]];
%t A321964 Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 07 2019, translated from Maple *)
%o A321964 (Sage) # uses[StieltjesGF from A321960]
%o A321964 def Trow(n, dim): return StieltjesGF(lambda k: n+k, dim, p=1)
%o A321964 for n in (0..7): print(Trow(n, 9))
%Y A321964 Rows of array: A000007, A001147, A000698, A167872, A321963.
%Y A321964 Columns include: A014105. Row sums of triangle: A321961.
%Y A321964 Cf. A321960.
%K A321964 nonn,tabl
%O A321964 0,8
%A A321964 _Peter Luschny_, Dec 26 2018