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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a142de360b77238e7125c5d76818a2c3ad6c0471096dfe71fdb160d2c2f2b807","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,     3;\n  2:    5,    15,     9;\n  3:   10,    45,    63,     27;\n  4:   25,   150,   315,    270,     81;\n  5:   50,   375,  1080,   1485,    972,    243;\n  6:  125,  1125,  4050,   7425,   7290,   3645,    729;\n  7:  250,  2625, 11475,  27000,  36855,  29160,  12393,   2187;\n  8:  625,  7500, 38250, 108000, 184275, 194400, 123930,  43740,   6561;\n  9: 1250, 16875, 99000, 330750, 692550, 941625, 831060, 459270, 144342, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+15*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[(5 + 15*x + 9*x^2)*P[n - 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