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
Examples
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;
Crossrefs
Programs
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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);
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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
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].