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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, ....
4

%I #26 Mar 24 2020 13:25:12

%S 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 0,10395,8162,2529,444,55,6,1,0,135135,110410,36243,6636,815,78,7,1,0,

%U 2027025,1708394,591381,114084,14425,1350,105,8,1

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

%H 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 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009.

%F 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 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 First few rows of the array start:

%e [0] 1, 0, 0, 0, 0, 0, 0, 0, ... A000007

%e [1] 1, 1, 3, 15, 105, 945, 10395, 135135, ... A001147

%e [2] 1, 2, 10, 74, 706, 8162, 110410, 1708394, ... A000698

%e [3] 1, 3, 21, 207, 2529, 36243, 591381, 10786527, ... A167872

%e [4] 1, 4, 36, 444, 6636, 114084, 2194596, 46460124, ... A321963

%e [5] 1, 5, 55, 815, 14425, 289925, 6444175, 155928575, ...

%e [6] 1, 6, 78, 1350, 27630, 636390, 16074990, 438572070, ...

%e Seen as triangle:

%e [0] 1;

%e [1] 0, 1;

%e [2] 0, 1, 1;

%e [3] 0, 3, 2, 1;

%e [4] 0, 15, 10, 3, 1;

%e [5] 0, 105, 74, 21, 4, 1;

%e [6] 0, 945, 706, 207, 36, 5, 1;

%e [7] 0, 10395, 8162, 2529, 444, 55, 6, 1;

%e [8] 0, 135135, 110410, 36243, 6636, 815, 78, 7, 1;

%p JacobiCF := proc(a, b, p:=2) local m, k;

%p m := 1;

%p for k from nops(a) by -1 to 1 do

%p m := 1 - b[k]*x - a[k]*x^p/m od;

%p return 1/m end:

%p JacobiGF := proc(a, b, p:=2) local cf, l, ser;

%p cf := JacobiCF(a, b, p);

%p l := min(nops(a), nops(b));

%p ser := series(cf, x, l);

%p seq(coeff(ser, x, n), n = 0..l-1) end:

%p JacobiSquare := proc(a, p:=2) local cf, ser;

%p cf := JacobiCF(a, a, p);

%p ser := series(cf, x, nops(a));

%p seq(coeff(ser, x, n), n = 0..nops(a)-1) end:

%p StieltjesGF := proc(a, p:=2) local z, cf, ser;

%p z := [seq(0, n = 1..nops(a))];

%p cf := JacobiCF(a, z, p);

%p ser := series(cf, x, nops(a));

%p seq(coeff(ser, x, n), n = 0..nops(a)-1) end:

%p s := n -> [seq(n+k, k = 0..9)]:

%p Trow := n -> StieltjesGF(s(n), 1):

%p for n from 0 to 6 do lprint(Trow(n)) od;

%t nmax = 9;

%t 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 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 JacobiSquare[a_, p_:2] := Module[{cf, ser}, cf = JacobiCF[a, a, p]; ser = Series[cf, {x, 0, Length[a]}]; CoefficientList[ser, x]];

%t 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 s[n_] := Table[n + k, {k, 0, nmax}];

%t Trow[0] = Table[Boole[k == 0], {k, 0, nmax}];

%t Trow[n_] := Trow[n] = StieltjesGF[s[n], 1] ;

%t T[n_, k_] := Trow[n][[k + 1]];

%t Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 07 2019, translated from Maple *)

%o (Sage) # uses[StieltjesGF from A321960]

%o def Trow(n, dim): return StieltjesGF(lambda k: n+k, dim, p=1)

%o for n in (0..7): print(Trow(n, 9))

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

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

%Y Cf. A321960.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Dec 26 2018