Central Trinomial Coefficient

The $n$th central binomial coefficient is defined as the coefficient of $x^n$ in the expansion of $(1+x+x^2)^n$. The first few are 1, 3, 7, 19, 51, 141, 393, ... (Sloane's A002426). This sequence cannot be expressed as a fixed number of hypergeometric terms (Petkovsek et al. 1996, p. 160). The Generating Function is given by

$$f(x)={1\over\sqrt{(1+x)(1-3x)}}=1+x+3x^2+7x^3+\ldots.$$

See also Central Binomial Coefficient

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, 1996.

Sloane, N. J. A. Sequence A002426/M2673 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.