A385896 Array read by ascending antidiagonals: A(n, k) = k! * [x^k] (1 - sin(n*x))^(-1/n) for n > 0, A(0, k) = 1.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 11, 16, 1, 1, 1, 5, 19, 57, 61, 1, 1, 1, 6, 29, 136, 361, 272, 1, 1, 1, 7, 41, 265, 1201, 2763, 1385, 1, 1, 1, 8, 55, 456, 3001, 13024, 24611, 7936, 1, 1, 1, 9, 71, 721, 6301, 42125, 165619, 250737, 50521, 1

Offset

0,9

Links

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

Formula

A(n, k) = T(k, k, n - k) where T(n, k, m) = T(n, k-1, m) + m * T(n-1, n-k-1, m) for k > 0, T(n, 0, m) = T(n-1, n-1, m), and T(0, 0, m) = 1.

Column n is a linear recurrence with kernel [(-1)^k*A135278(n, k), k = 0..n].

Example

Table starts:
  [0] 1, 1, 1,  1,    1,     1,      1, ... [A000012]
  [1] 1, 1, 2,  5,   16,    61,    272, ... [A000111]
  [2] 1, 1, 3, 11,   57,   361,   2763, ... [A001586]
  [3] 1, 1, 4, 19,  136,  1201,  13024, ... [A007788]
  [4] 1, 1, 5, 29,  265,  3001,  42125, ... [A144015]
  [5] 1, 1, 6, 41,  456,  6301, 108576, ... [A230134]
  [6] 1, 1, 7, 55,  721, 11761, 240247, ... [A227544]
  [7] 1, 1, 8, 71, 1072, 20161, 476288, ... [A235128]
  [8] 1, 1, 9, 89, 1521, 32401, 869049, ... [A230114]
     [A000027]  | [A187277] | [A385898].
            [A028387]   [A385897]
.
Seen as a triangle:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1, 1;
  [3] 1, 1, 2,  1;
  [4] 1, 1, 3,  5,   1;
  [5] 1, 1, 4, 11,  16,    1;
  [6] 1, 1, 5, 19,  57,   61,    1;
  [7] 1, 1, 6, 29, 136,  361,  272,    1;
  [8] 1, 1, 7, 41, 265, 1201, 2763, 1385, 1;

Maple

MAX := 16: ser := n -> series((1 - sin(n*x))^(-1/n), x, MAX):
A := (n, k) -> if n = 0 then 1 else k!*coeff(ser(n), x, k) fi:
seq(lprint(seq(A(n, k), k = 0..8)), n = 0..8);

Mathematica

T[n_, k_, m_] := T[n, k, m] =
  Which[
    n <  0 || k <  0, 0,
    n == 0 && k == 0, 1,
    k == 0, T[n - 1, n - 1, m],
    True, T[n, k - 1, m] + m*T[n - 1, n - k - 1, m]
];
A[n_, k_] := T[k, k, n - k];
Table[A[n, k], {n, 0, 10}, {k, 0, n}] // Flatten

Crossrefs

Rows: A000012, A000111, A001586, A007788, A144015, A230134, A227544, A235128, A230114.

Columns: A000027, A028387, A187277, A385897, A385898.

Cf. A135278.

Sequence in context: A299045 A124530 A243631 * A070914 A305962 A144150

Adjacent sequences: A385893 A385894 A385895 * A385897 A385898 A385899

Keyword

nonn,tabl

Author

Peter Luschny, Jul 20 2025

Status

approved