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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":"1ee54982a825c29dbec69b5a4d9251c883fdfb08ca8f976ef9f584a41b5932fb","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-25ac1cc25c118491","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.","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:           14,            14,             1;\n  3:          174,           244,            72,             1;\n  4:         3322,          6126,          3124,           320,            1;\n  5:        89756,        203316,        145384,         33156,         1334,            1;\n  6:      3211420,       8600272,       7917220,       2831220,       308270,         5418,          1;\n  7:    146132808,     450929052,     506282792,     246456720,     47567010,      2618624,      21788,         1;\n  8:   8202467544,   28599095824,   37804753460,   23287634672,   6569227270,    710553890,   20923420,     87308,      1;\n  9: 554489060200, 2152509769144, 3268840371744, 2433534378484, 907531806972, 154339736988, 9682642548, 160346280, 349434, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-1+(3*n-2)*x)*P(n-1,x) + (4+x-3*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[(3*n - 1 + (3*n - 2)*x)*P[n - 1, x] + (4 + x - 3*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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