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Bell Triangle

The Bell triangle, also called Aitken's array or the Peirce triangle (Knuth 2005, p. 28), is the number triangle obtained by beginning the first row with the number one, and beginning subsequent rows with last number of the previous row. Rows are filled out by adding the number in the preceding column to the number above it (OEIS A011971). The Bell numbers 1, 1, 2, 5, 15, ... (OEIS A000110) are then given as the values in the first column.The name "Bell triangle" was suggested to Gardner by J. Shallit. A reflected version is sometimes also considered (Knuth 2005, p. 28).The sums of the numbers in rows arewhere is a Stirling number of the second kind, giving the first few for , 2, ... as 1, 3, 10, 37, 151, ... (OEIS A005493).

Losanitsch's Triangle

(1)Losanitsch's triangle (OEIS A034851) is a number triangle for which each term is the sum of the two numbers immediately above it, except that, numbering the rows by , 1, 2, ... and the entries in each row by , 1, 2, ..., , are given by the recurrence equations(2)where is a binomial coefficient. can be written in closed form as(3)The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Losanitsch's triangle.The row sums of Losanitsch's triangle are(4)the first few terms of which are 1, 2, 3, 6, 10, 20, 36, ... (OEIS A005418).

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