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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":"0b78a05c7a8b32fa4989085cf9165ec69e215c23a5f0d59a9b61370c7aba6770","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-0b0f0dd4d2c8eb05","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2.","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:          11,           11,            1;\n  3:         103,          146,           45,            1;\n  4:        1442,         2677,         1393,          158,           1;\n  5:       28023,        63823,        46265,        10975,         512,           1;\n  6:      712805,      1917362,      1781640,       650940,       75450,        1593,         1;\n  7:    22857916,     70781135,     80015602,     39502740,     7876875,      471375,      4859,        1;\n  8:   898132469,   3140399343,   4173133419,   2596224062,   747179325,    84562626,   2755502,    14684,     1;\n  9: 42280116147, 164514396100, 250868890047, 188139393033, 71127493222, 12433552095, 826175938, 15378314, 44190, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+(2*n-1)*x)*P(n-1,x) + (3+x-2*x^2)*d/dx P(n-1,x) + (2+4*x+2*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[(2*n + (2*n - 1)*x)*P[n - 1, x] + (3 + x - 2*x^2)*D[P[n - 1, x], x] + (2 + 4*x + 2*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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