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

Author

Seiichi Manyama, Apr 18 2025

Keywords

• nonn
• tabl

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

Columns k=0..3 give A000007, A123227(n-1), A383163, A383164.

Row sums give A122704.

Cf. A001787, A178987, A383150, A383151, A383152, A383155.

Cf. A129062, A209849, A383140.

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).