This site is supported by donations to The OEIS Foundation.

Binomial transform

From OeisWiki
Jump to: navigation, search

This article needs more work.

Please help by expanding it!

The binomial transform is a bijective sequence transform based on convolution with binomial coefficients.


The binomial transform maps a sequence to and from a sequence via the 2-way mapping

Matrix interpretation

Considering the sequences a, b as column vectors/matrices A, B, these transformations can be written as multiplication with the lower left triangular infinite square matrix P which consists in Pascal's triangle of binomial coefficients A007318, and its inverse A130595 which just differs in the signs of every other element,


The binomial transform of the sequence (bn) = (1, b, b², ...) yields the sequence ((b+1)n) = (1, b+1, (b+1)², ...). The proof is easy, actually this is nothing else than Newton's binomial formula for (1+b)n.) This includes the following special cases:

  • b=1: The constant sequence A000012 = (1,1,1,1,...) transforms into the powers of 2, A000079 = (1, 2, 4, 8, 16,...).
  • b=-1: The sequence (-1)n = (1,-1,1,-1,...) transforms into the sequence (1,0,0,0,...) = (0^n).
  • b=2: The powers of 2, A000079 = (1, 2, 4, 8, 16,...), transform into powers of 3, A000244 = (1, 3, 9, 27, 81, 243, ...).

See also

External links


  • Daniel Forgues created this page on Aug 22 2010.
  • M. F. Hasler provided matrix formulae, references to Pascal's triangle, binomial coefficients, OEIS sequences, links and other sections, on Nov 04 2014.

Cite this page as

D. Forgues and M. F. Hasler, Binomial transform. — From the On-Line Encyclopedia of Integer Sequences® (OEIS®) wiki. (Available at