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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).

%I A383149 #35 Apr 18 2025 08:44:21
%S A383149 1,0,1,0,3,1,0,12,9,1,0,66,75,18,1,0,480,690,255,30,1,0,4368,7290,
%T A383149 3555,645,45,1,0,47712,88536,52290,12705,1365,63,1,0,608016,1223628,
%U A383149 831684,249585,36120,2562,84,1,0,8855040,19019664,14405580,5073012,915705,87696,4410,108,1
%N 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).
%H A383149 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RisingFactorial.html">Rising Factorial</a>
%F A383149 f_n(m) = (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).
%F A383149 T(n,k) = [m^k] f_n(-m).
%F A383149 T(n,k) = Sum_{j=k..n} 2^(n-j) * Stirling2(n,j) * |Stirling1(j,k)|.
%F A383149 T(n,k) = [x^k] Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * RisingFactorial(x,k).
%F A383149 Sum_{k=0..n} (-1)^k * T(n,k) = f_m(1) = -2^(n-1) for n > 0.
%F A383149 E.g.f. of column k (with leading zeros): g(x)^k / k! with g(x) = -log(1 - (exp(2*x) - 1)/2).
%e A383149 f_0(m) = 1.
%e A383149 f_1(m) =      -m.
%e A383149 f_2(m) =    -3*m +     m^2.
%e A383149 f_3(m) =   -12*m +   9*m^2 -     m^3.
%e A383149 f_4(m) =   -66*m +  75*m^2 -  18*m^3 +    m^4.
%e A383149 f_5(m) =  -480*m + 690*m^2 - 255*m^3 + 30*m^4 - m^5.
%e A383149 Triangle begins:
%e A383149   1;
%e A383149   0,     1;
%e A383149   0,     3,     1;
%e A383149   0,    12,     9,     1;
%e A383149   0,    66,    75,    18,     1;
%e A383149   0,   480,   690,   255,    30,    1;
%e A383149   0,  4368,  7290,  3555,   645,   45,  1;
%e A383149   0, 47712, 88536, 52290, 12705, 1365, 63, 1;
%e A383149   ...
%o A383149 (PARI) T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*abs(stirling(j, k, 1)));
%o A383149 (Sage)
%o A383149 def a_row(n):
%o A383149     s = sum(2^(n-k)*stirling_number2(n, k)*rising_factorial(x, k) for k in (0..n))
%o A383149     return expand(s).list()
%o A383149 for n in (0..9): print(a_row(n))
%Y A383149 Columns k=0..3 give A000007, A123227(n-1), A383163, A383164.
%Y A383149 Row sums give A122704.
%Y A383149 Cf. A001787, A178987, A383150, A383151, A383152, A383155.
%Y A383149 Cf. A129062, A209849, A383140.
%K A383149 nonn,tabl
%O A383149 0,5
%A A383149 _Seiichi Manyama_, Apr 18 2025