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

%I #13 Jul 13 2019 03:34:01

%S 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,

%T 426,303,116,30,6,1,0,877,2146,1752,744,205,42,7,1,0,4140,11624,10845,

%U 5140,1535,330,56,8,1,0,21147,67146,71139,37676,12300,2820,497,72,9,1

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

%C For definitions and comments see A321964.

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

%e First few rows of the array start:

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

%e [1] 1, 1, 2, 5, 15, 52, 203, 877, 4140, ... A000110

%e [2] 1, 2, 6, 22, 92, 426, 2146, 11624, 67146, ... A074664

%e [3] 1, 3, 12, 57, 303, 1752, 10845, 71139, 491064, ... A321959

%e [4] 1, 4, 20, 116, 744, 5140, 37676, 290224, 2334300, ...

%e [5] 1, 5, 30, 205, 1535, 12300, 103975, 918785, 8434740, ...

%e [6] 1, 6, 42, 330, 2820, 25662, 245358, 2443272, 25188870, ...

%e [7] 1, 7, 56, 497, 4767, 48496, 516761, 5719399, 65369136, ...

%e Seen as triangle:

%e [0] 1;

%e [1] 0, 1;

%e [2] 0, 1, 1;

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

%e [4] 0, 5, 6, 3, 1;

%e [5] 0, 15, 22, 12, 4, 1;

%e [6] 0, 52, 92, 57, 20, 5, 1;

%e [7] 0, 203, 426, 303, 116, 30, 6, 1;

%e [8] 0, 877, 2146, 1752, 744, 205, 42, 7, 1;

%p # The function JacobiSquare is defined in A321964.

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

%p for n from 0 to 7 do lprint(Trow(n)) od;

%t nmax = 10;

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

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

%t row[n_] := row[n] = JacobiSquare[s[n]];

%t T[_, 0] = 1; T[0, _] = 0; T[n_, k_] := row[n][[k + 1]];

%t Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jul 13 2019, after _Peter Luschny_ in A321964 *)

%o (Sage)

%o def JacobiCF(a, b, dim, p=2):

%o m = 1

%o for k in range(dim-1, -1, -1):

%o m = 1 - b(k)*x - a(k)*x^p/m

%o return 1/m

%o def JacobiGF(a, b, dim, p=2):

%o cf = JacobiCF(a, b, dim, p)

%o return cf.series(x, dim).list()

%o def JacobiSquare(a, dim, p=2):

%o cf = JacobiCF(a, a, dim, p)

%o return cf.series(x, dim).list()

%o def StieltjesGF(a, dim, p=2):

%o return JacobiGF(a, lambda n: 0, dim, p)

%o def Trow(n): return JacobiSquare(lambda k: n+k, 10)

%o for n in (0..4): print(Trow(n))

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

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

%Y Cf. A321964.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Dec 27 2018

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Last modified March 29 09:28 EDT 2024. Contains 371268 sequences. (Running on oeis4.)