A080839 -id:A080839 - cp's OEIS frontend

A363003 Number of integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).

1, 1, 2, 6, 26, 166, 1562, 21614, 438594, 13032614, 566069882

Offset: 0

Pontus von Brömssen, May 13 2023

a(n) is even for all n >= 2, because if the sequence (x_1, ..., x_n) has Gilbreath transform (1, ..., 1), so has the sequence (2 - x_1, ..., 2 - x_n).

Negative terms are permitted.

			For n = 4, the following 13 sequences, together with the sequences obtained by replacing each term x by 2-x in each of these sequences, have Gilbreath transform (1, 1, 1, 1), so a(4) = 26.
  (1, 2, 0, -4),
  (1, 2, 0, -2),
  (1, 2, 0,  0),
  (1, 2, 0,  2),
  (1, 2, 0,  4),
  (1, 2, 2,  0),
  (1, 2, 2,  2),
  (1, 2, 2,  4),
  (1, 2, 4,  0),
  (1, 2, 4,  2),
  (1, 2, 4,  4),
  (1, 2, 4,  6),
  (1, 2, 4,  8).

Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363004 (distinct positive integers), A363005 (distinct integers).

A363002 Number of positive nondecreasing integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 1, 2, 5, 17, 82, 573, 5839, 86921, 1890317, 60013894, 2778068147

Offset: 0

Pontus von Brömssen, May 13 2023

			For n = 4, the a(4) = 5 sequences are:
  (1, 2, 2, 2),
  (1, 2, 2, 4),
  (1, 2, 4, 4),
  (1, 2, 4, 6),
  (1, 2, 4, 8).

Cf. A080839 (increasing sequences), A362451, A363003, A363004 (distinct positive integers), A363005 (distinct integers).

A363004 Number of sequences of n distinct positive integers whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 40, 355, 4819, 99242, 3049155, 138762035

Offset: 0

Pontus von Brömssen, May 13 2023

			For n = 5, the a(5) = 7 sequences are:
  (1, 2, 4, 6,  8),
  (1, 2, 4, 6, 10),
  (1, 2, 4, 8,  6),
  (1, 2, 4, 8, 10),
  (1, 2, 4, 8, 12),
  (1, 2, 4, 8, 14),
  (1, 2, 4, 8, 16).

Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363003, A363005 (distinct integers).

A363005 Number of sequences of n distinct integers whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 2, 4, 12, 56, 416, 4764, 84272, 2278740, 92890636, 5659487836

Offset: 0

Pontus von Brömssen, May 13 2023

a(n) is even for all n >= 2, because if the sequence (x_1, ..., x_n) has Gilbreath transform (1, ..., 1), so has the sequence (2 - x_1, ..., 2 - x_n).

Negative terms are permitted.

			For n = 4, the following 6 sequences, together with the sequences obtained by replacing each term x by 2-x in each of these sequences, have Gilbreath transform (1, 1, 1, 1), so a(4) = 12.
  (1, 2, 0, -4),
  (1, 2, 0, -2),
  (1, 2, 0,  4),
  (1, 2, 4,  0),
  (1, 2, 4,  6),
  (1, 2, 4,  8).

Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363003, A363004 (distinct positive integers).

cp's OEIS Frontend

A363003 Number of integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).

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A363002 Number of positive nondecreasing integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).

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A363004 Number of sequences of n distinct positive integers whose Gilbreath transform is (1, 1, ..., 1).

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A363005 Number of sequences of n distinct integers whose Gilbreath transform is (1, 1, ..., 1).

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