A001236 -id:A001236 - cp's OEIS frontend

A008969 Triangle of differences of reciprocals of unity.

1, 1, 3, 1, 11, 7, 1, 50, 85, 15, 1, 274, 1660, 575, 31, 1, 1764, 48076, 46760, 3661, 63, 1, 13068, 1942416, 6998824, 1217776, 22631, 127, 1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255, 1, 1026576, 7245893376, 673781602752, 1413470290176, 117550462624, 747497920, 833375, 511

Offset: 1

N. J. A. Sloane

			Triangle T(n,k) begins:
  1;
  1,      3;
  1,     11,         7;
  1,     50,        85,         15;
  1,    274,      1660,        575,        31;
  1,   1764,     48076,      46760,      3661,       63;
  1,  13068,   1942416,    6998824,   1217776,    22631,    127;
  1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255;
  ...

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.

Alois P. Heinz, Rows n = 1..45, flattened

Cf. A001236, A001237, A001238, A001240, A001241, A001242.

Columns include A000254, A000424, A001236, A001237, A001238. Right-hand columns include A000225, A001240, A001241, A001242.

T:= (n,k)-> `if`(k<=n, (n-k+2)!^k *
     add((-1)^(j+1)*binomial(n-k+2, j)/ j^k, j=1..n-k+2), 0):
seq(seq(T(n,k), k=0..n), n=0..7); # Alois P. Heinz, Sep 05 2008

T[n_, k_] := If[k <= n, (n-k+2)!^k*Sum[(-1)^(j+1)*Binomial[n-k+2, j]/j^k, {j, 1, n-k+2}], 0]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

A384561 One fourth of the number of permutations of [n] with |p(i+1) - p(i)| >= 2, for i = 1..(n-1) and n appears at position i = 1 or i = n.

Original entry on oeis.org

1, 6, 39, 284, 2337, 21474, 218179, 2430216, 29459301, 386182478, 5444570631, 82157021556, 1321282006249, 22562446559034, 407722012334667, 7773697259015264, 155956589714240109, 3284208113313605286, 72434065593967762831, 1669777527837108720588, 40157785493048522566641

Offset: 5

Wolfdieter Lang, Jun 04 2025

The number of such permutations of [n] is 1 for n = 1 (the p(i) condition is not needed), and 0 for n = 2, 3 and 4, hence a(1) = 1/4 and a(n) = 0 for n = 2, 3 and 4.

The number of permutations of [n] with |p(i+1) - p(i)| >= 2, for i = 1..(n-1), for n >= n is given by A002464(n), for n >= 0. See also A001266(n) = A002464(n)/2, for n >= 2. These permutations are also called king permutations, e.g., in A382644.

			n = 5: the 4 permutations are 2 4 1 3 5, 3 1 4 2 5 and their reversals 5 3 1 4 2,  5 2 4 1 3.
a(5) = 1 = A382644(4)/2 = (A001236(5) - A242522(6))/2 = (7 - 5)/2, and A382644(5)/2 - A242522(6) = 6 - 5 = 1
a(6) = a(5) + A242522(6) = 1 + 5 = 6.

Cf. A002464, A001266, A242522, A382644.

a(n) = A382644(n-1)/2, for n >= 5.
a(n) = (A001266(n) - A242522(n+1))/2, for n >= 5.
a(n) = A382644(n)/2 - A242522(n+1), for n >= 5.
a(n) = a(n-1) + A242522(n), for n >= 6, with a(5) = 1.

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A008969 Triangle of differences of reciprocals of unity.

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