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Bernoulli Triangle

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(1)

The Bernoulli triangle is the number triangle illustrated above (OEIS A008949) composed of the partial sums of binomial coefficients,

a_(nk)=sum_(i=0)^(k)(n; i)
(2)
=2^n-(Gamma(n+1)_2F_1(1,k-n+1;k+2;-1))/(Gamma(k+2)Gamma(n-k)),
(3)

where Gamma(n) is a gamma function and _2F_1(a,b;c;a) is a hypergeometric function.

Binary plot of the Bernoulli triangle

The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Bernoulli triangle.

See also

Binomial Coefficient

Explore with Wolfram|Alpha

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More things to try:

  • Pascal's triangle
  • binomial coefficients
  • number triangles

References

Kirillov, A. A. "Variations on the Triangular Theme." In Lie Groups and Lie Algebras: E. B. Dynkin's Seminar: Dedicated to E. B. Dynkin on the Occasion of His Seventieth Birthday (Ed. S. G. Gindikin and E. B. Vinberg.) Providence, RI: Amer. Math. Soc., pp. 43-73, 1995.MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-Correcting Codes. Amsterdam, Netherlands: North-Holland, p. 376, 1978.Sloane, N. J. A. Sequence A008949 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Bernoulli Triangle

Cite this as:

Weisstein, Eric W. "Bernoulli Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BernoulliTriangle.html

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