Input
binomial(n, r) | 69 digits
Result
binomial(n, r)
Plots
n |
0 |
1 |
2 |
3 |
Alternate form
(Gamma(n + 1))/(Gamma(r + 1) Gamma(n – r + 1))
Roots
(no roots exist)
Series expansion at r=0
1 + r (polygamma(0, n + 1) + gamma) + 1/12 r^2 (6 polygamma(0, n + 1)^2 + 12 gamma polygamma(0, n + 1) – 6 polygamma(1, n + 1) – pi^2 + 6 gamma^2) + 1/12 r^3 (2 polygamma(0, n + 1)^3 + 6 gamma polygamma(0, n + 1)^2 + (-6 polygamma(1, n + 1) – pi^2 + 6 gamma^2) polygamma(0, n + 1) – 6 gamma polygamma(1, n + 1) + 2 polygamma(2, n + 1) – gamma pi^2 + 2 gamma^3 – 2 polygamma(2, 1)) + (r^4 (60 polygamma(0, n + 1)^4 + 240 gamma polygamma(0, n + 1)^3 + 60 (-6 polygamma(1, n + 1) – pi^2 + 6 gamma^2) polygamma(0, n + 1)^2 + 120 (-6 gamma polygamma(1, n + 1) + 2 polygamma(2, n + 1) – gamma pi^2 + 2 gamma^3 – 2 polygamma(2, 1)) polygamma(0, n + 1) + 180 polygamma(1, n + 1)^2 – 60 (6 gamma^2 – pi^2) polygamma(1, n + 1) + 240 gamma polygamma(2, n + 1) – 60 polygamma(3, n + 1) + pi^4 – 60 gamma^2 pi^2 + 60 gamma^4 – 240 gamma polygamma(2, 1)))/1440 + O(r^5)
(Taylor series)
Series expansion at r=∞
sin(pi (n – r + 1)) ((Gamma(n + 1) sqrt(1/r))/(sqrt(2) pi^(3/2)) – (Gamma(n + 1) (1/r)^(3/2))/(12 (sqrt(2) pi^(3/2))) + (Gamma(n + 1) (1/r)^(5/2))/(288 sqrt(2) pi^(3/2)) + (139 Gamma(n + 1) (1/r)^(7/2))/(51840 sqrt(2) pi^(3/2)) – (571 Gamma(n + 1) (1/r)^(9/2))/(2488320 (sqrt(2) pi^(3/2))) + O((1/r)^(11/2))) exp(1/2 (log(2 pi) – (2 n + 1) log(r)) + (6 n^2 + 6 n + 1)/(12 r) + (n (2 n^2 + 3 n + 1))/(12 r^2) + (30 n^4 + 60 n^3 + 30 n^2 – 1)/(360 r^3) + (n (6 n^4 + 15 n^3 + 10 n^2 – 1))/(120 r^4) + O((1/r)^5))
Derivative
(d)/(dr)(binomial(n, r) | 69 digits) = binomial(n, r) (polygamma(0, n – r + 1) – polygamma(0, r + 1))