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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"d71b0024c4ef52bc0bd4ee1ca97b80bf4bd3347a8743a58cbc2d36cead4e6582","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ea2139b8dbf6bf8d","role":"sequence","name":"Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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,  3;\n  2: 1,  6,   9;\n  3: 2, 15,  36,   27;\n  4: 1, 12,  54,  108,    81;\n  5: 2, 27, 144,  378,   486,   243;\n  6: 1, 18, 135,  540,  1215,  1458,   729;\n  7: 2, 39, 324, 1485,  4050,  6561,  5832,  2187;\n  8: 1, 24, 252, 1512,  5670, 13608, 20412, 17496,  6561;\n  9: 2, 51, 576, 3780, 15876, 44226, 81648, 96228, 65610, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+6*x+9*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 6*x + 9*x^2)*P[n - 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