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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"39be7da43247ace1eef2d6164cc2bbf4b2b9db1412b57514bd0cf32c08f0d122","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2b477f2aa77e3efc","role":"sequence","name":"Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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,          4;\n  2:         8,         40,          48;\n  3:        48,        400,        1088,          960;\n  4:       384,       4544,       19904,        38144,         26880;\n  5:      3840,      59264,      362624,      1097984,       1641984,        967680;\n  6:     46080,     880128,     6959104,     29131264,      68015104,      83859456,       42577920;\n  7:    645120,   14717952,   143194112,    769711104,    2467037184,    4710817792,     4956782592,     2214051840;\n  8:  10321920,  274194432,  3174182912,  20907024384,   85655261184,  223395315712,   361957588992,   332831784960,   132843110400;\n  9: 185794560, 5637390336, 75780513792, 592170876928, 2963472359424, 9845673443328, 21706118070272, 30604088180736, 25023737364480, 9033331507200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+(8*n-4)*x)*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 + (8*n - 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