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and c = 4.","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:    0;\n   2:    4;\n   3:   30,     20;\n   4:  108,    210,     60;\n   5:  280,   1092,    840,     140;\n   6:  600,   3920,   5880,    2520,     280;\n   7: 1134,  11160,  27720,   22680,    6300,     504;\n   8: 1960,  27090, 100800,  135240,   70560,   13860,    840;\n   9: 3168,  58520, 304920,  609840,  517440,  188496,  27720,  1320;\n  10: 4860, 115632, 803880, 2245320, 2827440, 1663200, 449064, 51480, 1980;\n","recurrence_sentence":"The row polynomials satisfy (n^4-4*n^3+3*n^2+4*n-4+(2*n^3-7*n^2+n+10)*x+(n^2-n-2)*x^2)*P(n,x) = (n^4-3*n^2+2*n+(3*n^4-8*n^3+5*n^2-2*n+4)*x+(6*n^3-17*n^2+17*n-10)*x^2+(3*n^2-3*n+2)*x^3)*P(n-1,x) + ((2*n^3-3*n^2-n+2)*x-(2*n^3-7*n^2+5*n+2)*x^2-(4*n^2-8*n)*x^3-2*n*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 - 3*n^2 + 2*n + (3*n^4 - 8*n^3 + 5*n^2 - 2*n + 4)*x + (6*n^3 - 17*n^2 + 17*n - 10)*x^2 + (3*n^2 - 3*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 - 3*n^2 - n + 2)*x - (2*n^3 - 7*n^2 + 5*n + 2)*x^2 - (4*n^2 - 8*n)*x^3 - 2*n*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^3 + 3*n^2 + 4*n - 4 + (2*n^3 - 7*n^2 + n + 10)*x + (n^2 - n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((n^4 - 3*n^2 + 2*n + (3*n^4 - 8*n^3 + 5*n^2 - 2*n + 4)*x + (6*n^3 - 17*n^2 + 17*n - 10)*x^2 + (3*n^2 - 3*n + 2)*x^3)*v[i - 1] + ((2*n^3 - 3*n^2 - n + 2)*x - (2*n^3 - 7*n^2 + 5*n + 2)*x^2 - (4*n^2 - 8*n)*x^3 - 2*n*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^3 + 3*n^2 + 4*n - 4 + (2*n^3 - 7*n^2 + n + 10)*x + (n^2 - n - 2)*x^2)); 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-a2304b34e998691c","steps":["D","D+1"],"edges":["e-36b6691ae4af4747","e-fbad4c5e2b273b38"],"text":"A062145 \u2192 D \u2192 D+1"},"headline":"A062145 \u2192 D \u2192 D+1","chain":null}
