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 A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n. 2
 1, 0, 1, 0, -2, 1, 0, 7, -6, 1, 0, -35, 40, -12, 1, 0, 228, -315, 130, -20, 1, 0, -1834, 2908, -1485, 320, -30, 1, 0, 17582, -30989, 18508, -5005, 665, -42, 1, 0, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS EXAMPLE Triangle starts: [0] [1] [1] [0,       1] [2] [0,      -2,        1] [3] [0,       7,       -6,       1] [4] [0,     -35,       40,     -12,        1] [5] [0,     228,     -315,     130,      -20,      1] [6] [0,   -1834,     2908,   -1485,      320,    -30,      1] [7] [0,   17582,   -30989,   18508,    -5005,    665,    -42,    1] [8] [0, -195866,   375611, -253400,    81088, -13650,   1232,  -56,   1] [9] [0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1] MATHEMATICA p[n_] := Sum[StirlingS1[n, k] FactorialPower[x, k] , {k, 0, n}]; Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten PROG (Sage) def a_row(n):     s = sum((-1)^(n-k)*stirling_number1(n, k)*falling_factorial(x, k) for k in (0..n))     return expand(s).list() [a_row(n) for n in (0..9)] CROSSREFS Cf. A039814 (variant), A129062, A325873. Sequence in context: A284797 A316135 A327620 * A021896 A188835 A217735 Adjacent sequences:  A325869 A325870 A325871 * A325873 A325874 A325875 KEYWORD sign,tabl AUTHOR Peter Luschny, Jun 27 2019 STATUS approved

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Last modified August 12 16:00 EDT 2020. Contains 336439 sequences. (Running on oeis4.)