Monotonic Ordinary Generating Functions

SPECIES

Simplicial Polytopic Numbers

$\displaystyle \dfrac{1}{1-x} = \sum_{n \geq 0} x^n = 1 + x + x^2 + x^3 + x^4 + \cdots$

$\displaystyle \dfrac{1}{(1-x)^2} = \sum_{n \geq 1} n \, x^{n-1} = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + \cdots$

$\displaystyle \dfrac{1}{(1-x)^3} = \sum_{n \geq 2} \dfrac{n(n-1)}{2} \, x^{n-2} = 1 + 3x + 6x^2 + 10x^3 + 15x^4 + \cdots$

$\displaystyle \dfrac{1}{(1-x)^4} = \sum_{n \geq 3} \dfrac{n(n-1)(n-2)}{6} \, x^{n-3} \\ = 1 + 4x + 10x^2 + 20x^3 + 35x^4 + \cdots$

$\displaystyle \dfrac{1}{(1-x)^5} = \sum_{n \geq 4} \dfrac{n(n-1)(n-2)(n-3)}{24} \, x^{n-4} \\ = 1 + 5x + 15x^2 + 35x^3 + 70x^4 + \cdots$

$\displaystyle \dfrac{1}{(1-x)^6} = \sum_{n \geq 5} \dfrac{n(n-1)(n-2)(n-3)(n-4)}{120} \, x^{n-5} \\ = 1 + 6x + 21x^2 + 56x^3 + 126x^4 + \cdots$

Square Roots

$\displaystyle \sqrt{1-x} = -\sum_{n \geq 0} \dfrac{(2n)!}{4^n\,(n!)^2\,(2n-1)} \, x^n = 1 - \dfrac{1}{2} x - \dfrac{1}{8} x^2 - \dfrac{1}{16} x^3 - \cdots$

$\displaystyle \dfrac{1}{\sqrt{1-x}} = \sum_{n \geq 0} \dfrac{(2n)!}{4^n\,(n!)^2} \, x^n = 1 + \dfrac{1}{2} x + \dfrac{3}{8} x^2 + \dfrac{5}{16} x^3 + \cdots$

Circular Trigonometric Functions

Hyperbolic Trigonometric Functions

$\displaystyle \sinh x = \sum_{n \geq 0} \dfrac{1}{(2n+1)!} \, x^{2n+1} = x + \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + \dfrac{x^7}{7!} + \cdots$

$\displaystyle \cosh x = \sum_{n \geq 0} \dfrac{1}{(2n)!} \, x^{2n} = 1 + \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + \dfrac{x^6}{6!} + \cdots$

$\displaystyle \tanh x = \sum_{n \geq 1} B_{2n} \, \dfrac{2^{2n}(2^{2n}-1)}{(2n)!} \, x^{2n-1} = x - \dfrac{x^3}{3} + \dfrac{2x^5}{15} - \dfrac{17x^7}{315} \pm \cdots$

$\displaystyle \coth x = \sum_{n \geq 0} B_{2n} \, \dfrac{2^{2n}}{(2n)!} \, x^{2n-1} = \dfrac{1}{x} + \dfrac{x}{3} - \dfrac{x^3}{45} + \dfrac{2x^5}{945} \pm \cdots$

$\displaystyle \text{sech}\,x = \sum_{n \geq 0} \dfrac{E_{2n}}{(2n)!} \, x^{2n} = 1 - \dfrac{x^2}{2} + \dfrac{5x^4}{24} - \dfrac{61x^6}{720} \pm \cdots$

$\displaystyle \text{csch}\,x = \sum_{n \geq 0} B_{2n} \dfrac{2(1-2^{2n-1})}{(2n)!} \, x^{2n-1} = \dfrac{1}{x} - \dfrac{x}{6} + \dfrac{7x^3}{360} - \dfrac{31x^5}{15120} \pm \cdots$

Inverse Hyperbolic Functions

$\displaystyle \text{artanh}\,x = \dfrac{1}{2} \ln\dfrac{1+x}{1-x} = \dfrac{\ln(1+x) - \ln(1-x)}{2} \\ \\ = \sum_{n \geq 0} \dfrac{x^{2n+1}}{2n+1} = x + \dfrac{x^3}{3} + \dfrac{x^5}{5} + \dfrac{x^7}{7} + \cdots$

$\displaystyle \text{arcoth}\,x = \text{artanh}\,\dfrac{1}{x} = \dfrac{1}{2} \ln\dfrac{x+1}{x-1} = \dfrac{\ln(x+1) - \ln(x-1)}{2} \\ \\ = \sum_{n \geq 0} \dfrac{x^{-(2n+1)}}{2n+1} = \dfrac{1}{x} + \dfrac{1}{3\,x^3} + \dfrac{1}{5\,x^5} + \dfrac{1}{7\,x^7} + \cdots$

$\displaystyle \ln(2x) - \text{arcosh}\,x = \ln(2x) + \ln(x-\sqrt{x^2-1}) \\ \\ = \sum_{n \geq 1} \dfrac{(2n)!}{2^{2n}\,(n!)^2} \, \dfrac{x^{-2n}}{2n} = \dfrac{1}{4\,x^2} + \dfrac{3}{32\,x^4} + \dfrac{5}{96\,x^6} + \dfrac{35}{1024\,x^8} + \cdots$

$\displaystyle \ln\dfrac{2}{x} - \text{arsech}\,x = \ln\dfrac{2}{x} - \text{arcosh}\,\dfrac{1}{x} = \ln\dfrac{2}{x} - \ln\dfrac{1+\sqrt{1-x^2}}{x} \\ = \ln\dfrac{2(1-\sqrt{1-x^2})}{x^2} = \ln2 - 2\ln x + \ln(1-\sqrt{1-x^2}) \\ \\ = \sum_{n \geq 1} \dfrac{(2n)!}{2^{2n}\,(n!)^2} \, \dfrac{x^{2n}}{2n} = \dfrac{x^2}{4} + \dfrac{4x^4}{32} + \dfrac{5x^6}{96} + \dfrac{35x^8}{1024} + \cdots$

Bernoulli Numbers

$\displaystyle \dfrac{1}{x} \; \psi^{(1)} \left(\dfrac{1}{x}\right) = \sum_{n \geq 0} B_n^{+} \, x^n = B_0 + B_1^{+} \, x + B_2 \, x^2 + B_3 \, x^3 + \cdots$

$\displaystyle -1 + \dfrac{1}{x} \; \psi^{(1)} \left(\dfrac{1}{x}\right) = \sum_{n \geq 1} B_n^{+} \, x^n = B_1^{+} \, x + B_2 \, x^2 + B_3 \, x^3 + \cdots$

$\displaystyle -x + \dfrac{1}{x} \; \psi^{(1)} \left(\dfrac{1}{x}\right) = \sum_{n \geq 0} B_n^{-} \, x^n = B_0 + B_1^{-} \, x + B_2 \, x^2 + B_3 \, x^3 + \cdots$

$\displaystyle -1 - x + \dfrac{1}{x} \; \psi^{(1)} \left(\dfrac{1}{x}\right) = \sum_{n \geq 1} B_n^{-} \, x^n = B_1^{-} \, x + B_2 \, x^2 + B_3 \, x^3 + \cdots$

$\displaystyle -\dfrac{x}{2} + \dfrac{1}{x} \; \psi^{(1)} \left(\dfrac{1}{x}\right) = \sum_{n \geq 0} B_{2n} \, x^{2n} = B_0 + B_2 \, x^2 + B_4 \, x^4 + B_6 \, x^6 + \cdots$

$\displaystyle -1 - \dfrac{x}{2} + \dfrac{1}{x} \; \psi^{(1)} \left(\dfrac{1}{x}\right) = \sum_{n \geq 1} B_{2n} \, x^{2n} = B_2 \, x^2 + B_4 \, x^4 + B_6 \, x^6 + \cdots$

$\displaystyle \dfrac{1}{2x^2} + \psi^{(1)}(1+x) = \sum_{n \geq 0} \dfrac{B_{2n}}{x^{2n+1}} = \dfrac{B_0}{x} + \dfrac{B_2}{x^3} + \dfrac{B_4}{x^5} + \dfrac{B_6}{x^7} + \cdots$

$\displaystyle -\dfrac{1}{x} + \dfrac{1}{2x^2} + \psi^{(1)}(1+x) = \sum_{n \geq 1} \dfrac{B_{2n}}{x^{2n+1}} = \dfrac{B_2}{x^3} + \dfrac{B_4}{x^5} + \dfrac{B_6}{x^7} + \cdots$

$\displaystyle \ln x + \dfrac{1}{2x} - \psi^{(0)}(1+x) = \sum_{n \geq 1} \dfrac{B_{2n}}{2n} \, x^{-2n} = \dfrac{B_2}{2x^2} + \dfrac{B_4}{4x^4} + \dfrac{B_6}{6x^6} + \cdots$

Catalan Numbers

$\displaystyle \dfrac{1-\sqrt{1-4x}}{2x} = \sum_{n \geq 0} \dfrac{1}{n+1} \, \binom{2n}{n} \, x^n \\ \\ = 1 + x + 2x^2 + 5x^3 + 14x^4 + 42x^5 + 132x^6 + \cdots$

Repeated Numbers

$\displaystyle \dfrac{1}{(1-x)(1-x^2)} = \dfrac{1}{(1+x)(1-x)^2} = \sum_{n \geq 1} \left\lfloor\dfrac{n+1}{2}\right\rfloor x^{n-1} \\ \\ = 1 + x + 2x^2 + 2x^3 + 3x^4 + 3x^5 + 4x^6 + 4x^7 + \cdots$