A321960 Array of sequences read by descending antidiagonals, A(n) the Jacobi square of the sequence n, n+1, n+2, ....

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 92, 57, 20, 5, 1, 0, 203, 426, 303, 116, 30, 6, 1, 0, 877, 2146, 1752, 744, 205, 42, 7, 1, 0, 4140, 11624, 10845, 5140, 1535, 330, 56, 8, 1, 0, 21147, 67146, 71139, 37676, 12300, 2820, 497, 72, 9, 1

Offset: 0

Author

Peter Luschny, Dec 27 2018

Keywords

• nonn
• tabl

Comments

For definitions and comments see A321964.

Examples

			First few rows of the array start:
[0] 1, 0,  0,   0,    0,     0,      0,       0,        0, ... A000007
[1] 1, 1,  2,   5,   15,    52,    203,     877,     4140, ... A000110
[2] 1, 2,  6,  22,   92,   426,   2146,   11624,    67146, ... A074664
[3] 1, 3, 12,  57,  303,  1752,  10845,   71139,   491064, ... A321959
[4] 1, 4, 20, 116,  744,  5140,  37676,  290224,  2334300, ...
[5] 1, 5, 30, 205, 1535, 12300, 103975,  918785,  8434740, ...
[6] 1, 6, 42, 330, 2820, 25662, 245358, 2443272, 25188870, ...
[7] 1, 7, 56, 497, 4767, 48496, 516761, 5719399, 65369136, ...
Seen as triangle:
[0] 1;
[1] 0,   1;
[2] 0,   1,    1;
[3] 0,   2,    2,    1;
[4] 0,   5,    6,    3,   1;
[5] 0,  15,   22,   12,   4,   1;
[6] 0,  52,   92,   57,  20,   5,  1;
[7] 0, 203,  426,  303, 116,  30,  6, 1;
[8] 0, 877, 2146, 1752, 744, 205, 42, 7, 1;
		

Crossrefs

Rows of array: A000007, A000110, A074664, A321959.

Columns include: A002378, A033445. Row sums of triangle: A321958.

Cf. A321964.

Programs

Maple

# The function JacobiSquare is defined in A321964.
s := n -> [seq(n+k, k = 0..9)]: Trow := n -> JacobiSquare(s(n)):
for n from 0 to 7 do lprint(Trow(n)) od;

Mathematica

nmax = 10;
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];
JacobiSquare[a_, p_: 2] := Module[{cf, ser}, cf = JacobiCF[a, a, p]; ser = Series[cf, {x, 0, Length[a]}]; CoefficientList[ser, x]];
s[n_] := Table[n + k, {k, 0, nmax}];
row[n_] := row[n] = JacobiSquare[s[n]];
T[, 0] = 1; T[0, ] = 0; T[n_, k_] := row[n][[k + 1]];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 13 2019, after Peter Luschny in A321964 *)

Sage

def JacobiCF(a, b, dim, p=2):
    m = 1
    for k in range(dim-1, -1, -1):
        m = 1 - b(k)*x - a(k)*x^p/m
    return 1/m
def JacobiGF(a, b, dim, p=2):
    cf = JacobiCF(a, b, dim, p)
    return cf.series(x, dim).list()
def JacobiSquare(a, dim, p=2):
    cf = JacobiCF(a, a, dim, p)
    return cf.series(x, dim).list()
def StieltjesGF(a, dim, p=2):
    return JacobiGF(a, lambda n: 0, dim, p)
def Trow(n): return JacobiSquare(lambda k: n+k, 10)
for n in (0..4): print(Trow(n))

Formula

T(n, k) = A(n)[k] where A(n) is the Jacobi square of the sequence s(j) = n + j, j >= 0.