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begins:\n   1: 1;\n   2: 1;\n   3: 1,   1;\n   4: 1,   4,     1;\n   5: 1,  11,    11,     1;\n   6: 1,  26,    66,    26,      1;\n   7: 1,  57,   302,   302,     57,     1;\n   8: 1, 120,  1191,  2416,   1191,   120,     1;\n   9: 1, 247,  4293, 15619,  15619,  4293,   247,   1;\n  10: 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 2)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, 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4293,  15619,   15619,    4293,    247,     1;\n   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;\n  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":2,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-efd1fdf07e5468d8","steps":[],"edges":[],"text":"A173018"},"headline":"A173018: Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","chain":null}