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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":"cbd2d859b6ef8c97f99af1df004c57f70cd733095912dcf3fd093fdf2fda6717","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-bdd5560e7eb65b04","role":"sequence","name":"Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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:          5,          3;\n   3:         34,         39,          9;\n   4:        287,        468,        216,         27;\n   5:       2839,       5901,       4023,       1026,         81;\n   6:      31898,      79863,      71190,      27405,       4455,        243;\n   7:     399079,    1161738,    1269324,     650700,     161595,      18225,       729;\n   8:    5480609,   18107085,   23304078,   14856912,    4972590,     862407,     71442,     2187;\n   9:   81724300,  301122975,  444378321,  337580082,  142040871,   33451866,   4276314,   271188,    6561;\n  10: 1311990425, 5320426626, 8829751194, 7763746563, 3929213043, 1172133585, 204405768, 20037294, 1003833, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) + (3+3*x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + 3*x)*P[n - 1, x] + (3 + 3*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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