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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"9d4d9d2298e2533d0d0e1ec385de02e4019287ffc7cf547c680dfbbc6507f11c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-f0c4b6406e0da4a3","role":"sequence","name":"Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-f0c4b6406e0da4a3","role":"sequence","name":"Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"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,     1;\n   2:   -1,    -1,     1;\n   3:    2,    -2,    -2,    1;\n   4:    0,     8,    -3,   -3,    1;\n   5:   -8,    -2,    20,   -4,   -4,     1;\n   6:   10,   -46,   -10,   40,   -5,    -5,    1;\n   7:   36,    76,  -156,  -30,   70,    -6,   -6,   1;\n   8: -112,   272,   322, -406,  -70,   112,   -7,  -7,  1;\n   9: -160, -1028,  1168, 1008, -896,  -140,  168,  -8, -8,  1;\n  10: 1188, -1468, -5220, 3744, 2604, -1764, -252, 240, -9, -9, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) - d/dx P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x] - 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] = x; for(n = 2, N, my(i = n); v[i] = -(1 - x)*v[i - 1] - 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-b0b0e9fb10acb2e2","steps":["x \u21a6 x\u22121 (shift -1)","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-9c6e37fd70f39446","e-7d57ae2f42078672"],"text":"A066325 \u2192 x \u21a6 x\u22121 (shift -1) \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A066325 \u2192 x \u21a6 x\u22121 (shift -1) \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}