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b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"3419f13b60ed8428e56a42f0bdd43a294c2bb28a1424a1d404ef295824e06263","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fc18ad0ea0731402","role":"sequence","name":"Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).","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: 71, 23, 1;\n 1: 729, 430, 45, 1;\n 2: 8842, 7809, 1441, 74, 1;\n 3: 125399, 146151, 41065, 3615, 110, 1;\n 4: 2026249, 2892574, 1125496, 148980, 7610, 153, 1;\n 5: 36458010, 61171803, 30782066, 5658996, 428295, 14231, 203, 1;\n 6: 719866701, 1384983979, 857216171, 206885210, 21575701, 1048754, 24430, 260, 1;\n 7: 15453821461, 33495652780, 24588329151, 7461006889, 1020316836, 67698183, 2282994, 39306, 324, 1;\n 8: 358100141148, 862125221275, 731120038771, 269526400340, 46561849390, 4027224453, 183773877, 4541580, 60105, 395, 1;\n 9: 8899677678109, 23525659003067, 22606820481377, 9851117431551, 2086705834235, 227451003703, 13438455723, 446009817, 8413185, 88220, 473, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+x)*P(n-1,x) + (7+7*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 71 + 23*x + x^2.","mathematica":"P[0, x_] := 71 + 23*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(8 + x)*P[n - 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