A383140 Triangle read by rows: the coefficients of polynomials (1/3^(m-n)) * Sum_{k=0..m} k^n * 2^(m-k) * binomial(m,k) in the variable m.

1, 0, 1, 0, 2, 1, 0, 2, 6, 1, 0, -6, 20, 12, 1, 0, -30, 10, 80, 20, 1, 0, 42, -320, 270, 220, 30, 1, 0, 882, -1386, -770, 1470, 490, 42, 1, 0, 954, 7308, -15064, 2800, 5180, 952, 56, 1, 0, -39870, 101826, -39340, -61992, 29820, 14364, 1680, 72, 1, 0, -203958, -40680, 841770, -666820, -86940, 139440, 34020, 2760, 90, 1

Offset: 0

Author

Seiichi Manyama, Apr 17 2025

Keywords

• sign
• tabl

Examples

			f_n(m) = (1/3^(m-n)) * Sum_{k=0..m} k^n * 2^(m-k) * binomial(m,k).
f_0(m) = 1.
f_1(m) =    m.
f_2(m) =  2*m +   m^2.
f_3(m) =  2*m + 6*m^2 + m^3.
Triangle begins:
  1;
  0,   1;
  0,   2,    1;
  0,   2,    6,   1;
  0,  -6,   20,  12,   1;
  0, -30,   10,  80,  20,  1;
  0,  42, -320, 270, 220, 30, 1;
  ...
		

Links

• Eric Weisstein's World of Mathematics, Falling Factorial

Crossrefs

Columns k=0..1 give A000007, A179929(n-1).

Row sums give A133494.

Alternating row sums give A212846.

Cf. A027471, A383136, A383137, A383138, A383139.

Cf. A129062, A209849.

Programs

PARI

T(n, k) = sum(j=k, n, 3^(n-j)*stirling(n, j, 2)*stirling(j, k, 1));

Sage

def a_row(n):
    s = sum(3^(n-k)*stirling_number2(n, k)*falling_factorial(x, k) for k in (0..n))
    return expand(s).list()
for n in (0..10): print(a_row(n))

Formula

T(n,k) = Sum_{j=k..n} 3^(n-j) * Stirling2(n,j) * Stirling1(j,k).

T(n,k) = [x^k] Sum_{k=0..n} 3^(n-k) * Stirling2(n,k) * FallingFactorial(x,k).

E.g.f. of column k (with leading zeros): g(x)^k / k! with g(x) = log(1 + (exp(3*x) - 1)/3).