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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a35575c6b4e4892ca70bc89d5a9e7dee7377cbbc2bd04b640e0496465a69449d","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-4e0b19699f4cf0db","role":"sequence","name":"Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= 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,     2;\n  2:    5,    10,     4;\n  3:   10,    30,    28,     8;\n  4:   25,   100,   140,    80,     16;\n  5:   50,   250,   480,   440,    192,     32;\n  6:  125,   750,  1800,  2200,   1440,    480,    64;\n  7:  250,  1750,  5100,  8000,   7280,   3840,  1088,   128;\n  8:  625,  5000, 17000, 32000,  36400,  25600, 10880,  2560,  256;\n  9: 1250, 11250, 44000, 98000, 136800, 124000, 72960, 26880, 5632, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+10*x+4*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(5 + 10*x + 4*x^2)*P[n - 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