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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":"d358de94ce8e9f4cc053383c96cc078a542f3a7f8a16db217d08a5ac7e5bd9b2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-56eee146d6109546","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.","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:           1;\n  1:           2,            1;\n  2:          12,           12,            1;\n  3:         126,          168,           44,            1;\n  4:        1902,         3258,         1482,          126,           1;\n  5:       39852,        83184,        52476,         9456,         314,          1;\n  6:     1092924,      2666952,      2144796,       618348,       48300,        720,         1;\n  7:    37613880,    104926212,    102606408,     40756248,     5672682,     212076,      1568,       1;\n  8:  1583720640,   4966044264,   5704697628,   2884702896,   604458090,   42964686,    838794,    3306,    1;\n  9: 79861657752, 277581544920, 365584940064, 224580965724, 63722645676, 7285531680, 281874096, 3082092, 6830, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+n*x)*P(n-1,x) + (6+5*x-x^2)*d/dx P(n-1,x) + (3+6*x+3*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 1 + n*x)*P[n - 1, x] + (6 + 5*x - x^2)*D[P[n - 1, x], x] + (3 + 6*x + 3*x^2)*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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