A383149 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^k * [m^k] (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).
1, 0, 1, 0, 3, 1, 0, 12, 9, 1, 0, 66, 75, 18, 1, 0, 480, 690, 255, 30, 1, 0, 4368, 7290, 3555, 645, 45, 1, 0, 47712, 88536, 52290, 12705, 1365, 63, 1, 0, 608016, 1223628, 831684, 249585, 36120, 2562, 84, 1, 0, 8855040, 19019664, 14405580, 5073012, 915705, 87696, 4410, 108, 1
Offset: 0
Examples
f_0(m) = 1. f_1(m) = -m. f_2(m) = -3*m + m^2. f_3(m) = -12*m + 9*m^2 - m^3. f_4(m) = -66*m + 75*m^2 - 18*m^3 + m^4. f_5(m) = -480*m + 690*m^2 - 255*m^3 + 30*m^4 - m^5. Triangle begins: 1; 0, 1; 0, 3, 1; 0, 12, 9, 1; 0, 66, 75, 18, 1; 0, 480, 690, 255, 30, 1; 0, 4368, 7290, 3555, 645, 45, 1; 0, 47712, 88536, 52290, 12705, 1365, 63, 1; ...
Links
- Eric Weisstein's World of Mathematics, Rising Factorial
Crossrefs
Programs
-
PARI
T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*abs(stirling(j, k, 1)));
-
Sage
def a_row(n): s = sum(2^(n-k)*stirling_number2(n, k)*rising_factorial(x, k) for k in (0..n)) return expand(s).list() for n in (0..9): print(a_row(n))
Formula
f_n(m) = (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).
T(n,k) = [m^k] f_n(-m).
T(n,k) = Sum_{j=k..n} 2^(n-j) * Stirling2(n,j) * |Stirling1(j,k)|.
T(n,k) = [x^k] Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * RisingFactorial(x,k).
Sum_{k=0..n} (-1)^k * T(n,k) = f_m(1) = -2^(n-1) for n > 0.
E.g.f. of column k (with leading zeros): g(x)^k / k! with g(x) = -log(1 - (exp(2*x) - 1)/2).