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Triangle read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,         4;\n   3:      2,        24,         9;\n   4:      6,       152,       135,         16;\n   5:     24,      1088,      1692,        448,        25;\n   6:    120,      8800,     21060,       9280,      1125,        36;\n   7:    720,     79680,    272520,     176640,     34750,      2376,       49;\n   8:   5040,    799680,   3722040,    3310720,    936250,    102312,     4459,     64;\n   9:  40320,   8816640,  53907840,   62899200,  23898000,   3701376,   255192,   7680,    81;\n  10: 362880, 105960960, 828455040, 1228021760, 600138000, 123246144, 11928168, 563712, 12393, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+10+(4*n-8)*x-9*x^2)*P(n-2,x) - ((6*n-12)*x-(4*n-8)*x^2-4*x^3)*d/dx P(n-2,x) - (9*x^2-12*x^3)*d^2/dx^2 P(n-2,x) + (2*n-6+4*x)*P(n-1,x) + (6*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 10 + (4*n - 8)*x - 9*x^2)*P[n - 2, x] - ((6*n - 12)*x - (4*n - 8)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (9*x^2 - 12*x^3)*D[P[n - 2, x], {x, 2}] + (2*n - 6 + 4*x)*P[n - 1, x] + (6*x - 4*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; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 7*n + 10 + (4*n - 8)*x - 9*x^2)*v[i - 2] - ((6*n - 12)*x - (4*n - 8)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (9*x^2 - 12*x^3)*deriv(deriv(v[i - 2], x), x) + (2*n - 6 + 4*x)*v[i - 1] + (6*x - 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e9164c616dcc1bbd","steps":["(\u03b8+1)\u00b2"],"edges":["e-5a303bfa5b5628dc"],"text":"A322944 \u2192 (\u03b8+1)\u00b2"},"headline":"A322944 \u2192 (\u03b8+1)\u00b2","chain":null}