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 are

where  is a Stirling number of the second kind, giving the first few for , 2, ... as 1, 3, 10, 37, 151, ... (OEIS A005493).

REFERENCES:

Aitken, A. C. "A Problem on Combinations." Edinburgh Math. Notes 28, 18-33, 1933.

Allouche, J.-P. and Shallit, J. Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, p. 205, 2003.

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 212, 1974.

Knuth, D. E. §7.2.1.4 in The Art of Computer Programming, Vol. 4: Combinatorial Algorithms. Fascicle 2: Generating All Tuples and Permutations. Reading, MA: Addison-Wesley, 2005.

Gardner, M. "The Tinkly Temple Bells." Ch. 2 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 24-38, 1992.

Peirce, C. S. "On the Algebra of Logic." Amer. J. Math. 3, 15-57, 1880. Reprinted in Collected Papers (1935-1958). Also reprinted in Writings of Charles S. Peirce: A Chronological Edition. Bloomington, IN: Indiana University Press, 1986.

Sloane, N. J. A. Sequences A000110/M1484, A005493, and A011971 in "The On-Line Encyclopedia of Integer Sequences."