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Triangle read by rows, for n >= 0 and 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:     0;\n  1:     2;\n  2:     2,      8;\n  3:     4,     24,      24;\n  4:    12,     88,     144,      64;\n  5:    48,    400,     840,     640,     160;\n  6:   240,   2192,    5400,    5440,    2400,     384;\n  7:  1440,  14112,   38976,   47040,   28000,    8064,    896;\n  8: 10080, 104544,  315168,  433216,  313600,  123648,  25088,  2048;\n  9: 80640, 876672, 2834976, 4306176, 3591840, 1741824, 489216, 73728, 4608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-4*n+4+(4*n-8)*x+4*x^2)*P(n-2,x) + (2*n-3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4 + (4*n - 8)*x + 4*x^2)*P[n - 2, x] + (2*n - 3 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 4 + (4*n - 8)*x + 4*x^2)*v[i - 2] + (2*n - 3 + 4*x)*v[i - 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-60c7e36cbe887b7e","steps":["D"],"edges":["e-f2695afea9239794"],"text":"A137320 \u2192 D"},"headline":"A137320 \u2192 D","chain":null}