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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"6e2846d11dac4cade0a077355dd1a5a9c4c8d35e462c509bf6b332a549af1565","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-590ad09c6151a6a8","role":"sequence","name":"basis_map_A013611 of Pascal-(1,5,1) array.","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:         6,         12;\n  2:        61,        204,         144;\n  3:       516,       2616,        4032,        1728;\n  4:      4621,      30648,       71424,       67392,        20736;\n  5:     40626,     335700,     1054080,     1546560,      1036800,       248832;\n  6:    359281,    3545172,    13977360,    27898560,     29341440,     15178752,      2985984;\n  7:   3171336,   36412464,   172900224,   437028480,    629544960,    511598592,    214990848,     35831808;\n  8:  28010041,  366716976,  2036495232,  6233079168,  11429061120,  12764086272,   8414502912,   2974040064,   429981696;\n  9: 247343646, 3637014108, 23126828544, 83136143232, 185331649536, 264295913472, 239726739456, 132613521408, 40418279424, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (25+60*x)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 12*x;\nP[n_, x_] := P[n, x] = Expand[(25 + 60*x)*P[n - 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