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Triangle read by rows, E2(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0, 2;\n  2: 0, 2,    6;\n  3: 0, 2,   24,     24;\n  4: 0, 2,   66,    232,     120;\n  5: 0, 2,  156,   1312,    2220,      720;\n  6: 0, 2,  342,   5808,   22000,    22248,     5040;\n  7: 0, 2,  720,  22440,  160600,   348840,   237888,    40320;\n  8: 0, 2, 1482,  79800,  979000,  3864120,  5497128,  2729088,   362880;\n  9: 0, 2, 3012, 269040, 5312500, 34593000, 87080448, 88210368, 33605280, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = ((4*n^2-6*n)*x-(4*n^3-10*n^2+6*n)*x^2)*P(n-2,x) + (n*x-(2*n^2-n)*x^2+(2*n^2-2*n)*x^3)*d/dx P(n-2,x) - (n-(4*n^2-6*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((4*n^2 - 6*n)*x - (4*n^3 - 10*n^2 + 6*n)*x^2)*P[n - 2, x] + (n*x - (2*n^2 - n)*x^2 + (2*n^2 - 2*n)*x^3)*D[P[n - 2, x], x] - (n - (4*n^2 - 6*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (((4*n^2 - 6*n)*x - (4*n^3 - 10*n^2 + 6*n)*x^2)*v[i - 2] + (n*x - (2*n^2 - n)*x^2 + (2*n^2 - 2*n)*x^3)*deriv(v[i - 2], x) - (n - (4*n^2 - 6*n + 2)*x)*v[i - 1] + ((n - 1)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e14e37227fef094e","steps":["reverse","\u03b8+1"],"edges":["e-c8375d3b3fe52ed3","e-5cb4b748e82d5be2"],"text":"A163936 \u2192 reverse \u2192 \u03b8+1"},"headline":"A163936 \u2192 reverse \u2192 \u03b8+1","chain":null}