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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"b84ac713a49eae3f2246eed5ade4842d1296bb08f976c5bd701a3276b4cf21e7","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-54284d24a094e2d3","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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":[{"id":"l-6537fa48e4d61274","kind":"oeis_match","node_a_id":"n-c5818ec04781ce60","node_b_id":null,"fingerprint":null,"oeis_id":"A375041","in_catalog":0,"target":"reversed_triangle","row_alignment":"node_lengths","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":8,"target":"reversed_triangle","terms_compared":46},"in_catalog":false,"oeis_id":"A375041","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A375041","relation":"match_reversed_triangle"}],"bfiles":[{"node_id":"n-c5818ec04781ce60","oeis_id":"A375041","target":"reversed_triangle","oeis_offset":1,"first_row":1,"rows_used":43,"terms_matched":46,"oeis_terms_agreeing":46,"oeis_terms_available":46,"term_count":989,"raw_size":43152}],"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:          1,           1;\n  2:          2,           3,          1;\n  3:         10,          17,          8,          1;\n  4:        100,         180,         97,         18,         1;\n  5:       1700,        3160,       1829,        403,        35,        1;\n  6:      44200,       83860,      50714,      12307,      1313,       61,      1;\n  7:    1635400,     3147020,    1960278,     506073,     60888,     3570,     98,     1;\n  8:   81770000,   158986400,  101160920,   27263928,   3550473,   239388,   8470,   148,   1;\n  9: 5315050000, 10415886000, 6734446200, 1873316240, 258044673, 19110693, 789938, 18090, 213, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-2*n+2+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[(n^2 - 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