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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.

%I A383140 #36 Apr 18 2025 08:44:25
%S A383140 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,
%T A383140 30,1,0,882,-1386,-770,1470,490,42,1,0,954,7308,-15064,2800,5180,952,
%U A383140 56,1,0,-39870,101826,-39340,-61992,29820,14364,1680,72,1,0,-203958,-40680,841770,-666820,-86940,139440,34020,2760,90,1
%N 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.
%H A383140 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FallingFactorial.html">Falling Factorial</a>
%F A383140 T(n,k) = Sum_{j=k..n} 3^(n-j) * Stirling2(n,j) * Stirling1(j,k).
%F A383140 T(n,k) = [x^k] Sum_{k=0..n} 3^(n-k) * Stirling2(n,k) * FallingFactorial(x,k).
%F A383140 E.g.f. of column k (with leading zeros): g(x)^k / k! with g(x) = log(1 + (exp(3*x) - 1)/3).
%e A383140 f_n(m) = (1/3^(m-n)) * Sum_{k=0..m} k^n * 2^(m-k) * binomial(m,k).
%e A383140 f_0(m) = 1.
%e A383140 f_1(m) =    m.
%e A383140 f_2(m) =  2*m +   m^2.
%e A383140 f_3(m) =  2*m + 6*m^2 + m^3.
%e A383140 Triangle begins:
%e A383140   1;
%e A383140   0,   1;
%e A383140   0,   2,    1;
%e A383140   0,   2,    6,   1;
%e A383140   0,  -6,   20,  12,   1;
%e A383140   0, -30,   10,  80,  20,  1;
%e A383140   0,  42, -320, 270, 220, 30, 1;
%e A383140   ...
%o A383140 (PARI) T(n, k) = sum(j=k, n, 3^(n-j)*stirling(n, j, 2)*stirling(j, k, 1));
%o A383140 (Sage)
%o A383140 def a_row(n):
%o A383140     s = sum(3^(n-k)*stirling_number2(n, k)*falling_factorial(x, k) for k in (0..n))
%o A383140     return expand(s).list()
%o A383140 for n in (0..10): print(a_row(n))
%Y A383140 Columns k=0..1 give A000007, A179929(n-1).
%Y A383140 Row sums give A133494.
%Y A383140 Alternating row sums give A212846.
%Y A383140 Cf. A027471, A383136, A383137, A383138, A383139.
%Y A383140 Cf. A129062, A209849.
%K A383140 sign,tabl
%O A383140 0,5
%A A383140 _Seiichi Manyama_, Apr 17 2025