login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A321960 Array of sequences read by descending antidiagonals, A(n) the Jacobi square of the sequence n, n+1, n+2, .... 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

For definitions and comments see A321964.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

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

EXAMPLE

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;

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 *)

PROG

(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))

CROSSREFS

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

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

Cf. A321964.

Sequence in context: A120059 A067347 A120568 * A189233 A242153 A065066

Adjacent sequences:  A321957 A321958 A321959 * A321961 A321962 A321963

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Dec 27 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 19 09:58 EDT 2021. Contains 348074 sequences. (Running on oeis4.)