A292068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + j^k*x^j).

1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -3, -2, 1, 1, -1, -7, -6, 2, -1, 1, -1, -15, -20, 6, -1, 1, 1, -1, -31, -66, 20, 5, 4, -1, 1, -1, -63, -212, 66, 71, 40, -1, 2, 1, -1, -127, -666, 212, 605, 442, 11, 18, -2, 1, -1, -255, -2060, 666, 4439, 4660, 215, 226, -22, 2

Offset

0,13

Links

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

Example

Square array begins:
    1,  1,  1,   1,   1, ...
   -1, -1, -1,  -1,  -1, ...
    0, -1, -3,  -7, -15, ...
   -1, -2, -6, -20, -66, ...
    1,  2,  6,  20,  66, ...

Maple

b:= proc(n, i, k) option remember; (m->
      `if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
      `if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(b(n-i$2, k)*A(i, k), i=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 12 2017

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[# < n, 0, If[n == #, i!^k, b[n, i-1, k] + If[i > n, 0, i^k b[n-i, i-1, k]]]]&[i(i+1)/2];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k] A[i, k], {i, 0, n-1}]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Nov 20 2019, after Alois P. Heinz *)

Prog

(Python)
from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, k):
    m=i*(i + 1)/2
    return 0 if m<n else f(i)**k if n==m else b(n, i - 1, k) + (0 if i>n else i**k*b(n - i, i - 1, k))
@cacheit
def A(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*A(i, k) for i in range(n)])
for d in range(13): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Sep 14 2017, after Maple program

Crossrefs

Columns k=0..2 give A081362, A022693, A292165.

Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000225.

Main diagonal gives A292072.

Cf. A292189, A292193.

Sequence in context: A027082 A140736 A284993 * A350942 A140056 A240239

Adjacent sequences: A292065 A292066 A292067 * A292069 A292070 A292071

Keyword

sign,tabl

Author

Seiichi Manyama, Sep 12 2017

Status

approved