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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"de985fb4ebad6c952af9c5e48188ff6094e5265fea02684b02f41859f9b1a023","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-e1d1088f7d1ff894","role":"sequence","name":"Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","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:           21,             30;\n  2:          441,           1260,            900;\n  3:         9261,          39690,          56700,           27000;\n  4:       194481,        1111320,        2381400,         2268000,          810000;\n  5:      4084101,       29172150,       83349000,       119070000,        85050000,        24300000;\n  6:     85766121,      735138180,     2625493500,      5000940000,      5358150000,      3061800000,       729000000;\n  7:   1801088541,    18010885410,    77189508900,    183784545000,    262549350000,    225042300000,    107163000000,     21870000000;\n  8:  37822859361,   432261249840,  2161306249200,   6175160712000,  11027072700000,  12602368800000,   9001692000000,   3674160000000,    656100000000;\n  9: 794280046581, 10212172027470, 58355268728400, 194517562428000, 416823348060000, 595461925800000, 567106596000000, 347208120000000, 124002900000000, 19683000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (21+30*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[(21 + 30*x)*P[n - 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