cp's OEIS Frontend

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.

%I A385896 #15 Jul 24 2025 11:51:53
%S A385896 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,
%T A385896 29,136,361,272,1,1,1,7,41,265,1201,2763,1385,1,1,1,8,55,456,3001,
%U A385896 13024,24611,7936,1,1,1,9,71,721,6301,42125,165619,250737,50521,1
%N 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.
%F A385896 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.
%F A385896 Column n is a linear recurrence with kernel [(-1)^k*A135278(n, k), k = 0..n].
%e A385896 Table starts:
%e A385896   [0] 1, 1, 1,  1,    1,     1,      1, ... [A000012]
%e A385896   [1] 1, 1, 2,  5,   16,    61,    272, ... [A000111]
%e A385896   [2] 1, 1, 3, 11,   57,   361,   2763, ... [A001586]
%e A385896   [3] 1, 1, 4, 19,  136,  1201,  13024, ... [A007788]
%e A385896   [4] 1, 1, 5, 29,  265,  3001,  42125, ... [A144015]
%e A385896   [5] 1, 1, 6, 41,  456,  6301, 108576, ... [A230134]
%e A385896   [6] 1, 1, 7, 55,  721, 11761, 240247, ... [A227544]
%e A385896   [7] 1, 1, 8, 71, 1072, 20161, 476288, ... [A235128]
%e A385896   [8] 1, 1, 9, 89, 1521, 32401, 869049, ... [A230114]
%e A385896      [A000027]  | [A187277] | [A385898].
%e A385896             [A028387]   [A385897]
%e A385896 .
%e A385896 Seen as a triangle:
%e A385896   [0] 1;
%e A385896   [1] 1, 1;
%e A385896   [2] 1, 1, 1;
%e A385896   [3] 1, 1, 2,  1;
%e A385896   [4] 1, 1, 3,  5,   1;
%e A385896   [5] 1, 1, 4, 11,  16,    1;
%e A385896   [6] 1, 1, 5, 19,  57,   61,    1;
%e A385896   [7] 1, 1, 6, 29, 136,  361,  272,    1;
%e A385896   [8] 1, 1, 7, 41, 265, 1201, 2763, 1385, 1;
%p A385896 MAX := 16: ser := n -> series((1 - sin(n*x))^(-1/n), x, MAX):
%p A385896 A := (n, k) -> if n = 0 then 1 else k!*coeff(ser(n), x, k) fi:
%p A385896 seq(lprint(seq(A(n, k), k = 0..8)), n = 0..8);
%t A385896 T[n_, k_, m_] := T[n, k, m] =
%t A385896   Which[
%t A385896     n <  0 || k <  0, 0,
%t A385896     n == 0 && k == 0, 1,
%t A385896     k == 0, T[n - 1, n - 1, m],
%t A385896     True, T[n, k - 1, m] + m*T[n - 1, n - k - 1, m]
%t A385896 ];
%t A385896 A[n_, k_] := T[k, k, n - k];
%t A385896 Table[A[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
%Y A385896 Rows: A000012, A000111, A001586, A007788, A144015, A230134, A227544, A235128, A230114.
%Y A385896 Columns: A000027, A028387, A187277, A385897, A385898.
%Y A385896 Cf. A135278.
%K A385896 nonn,tabl
%O A385896 0,9
%A A385896 _Peter Luschny_, Jul 20 2025