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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"799fda8afee4a4c65db0d50c5aadd013383aa29fd6ae14a88fb6906123ca6f86","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-30eb228e35a8e7fa","kind":"oeis_match","node_a_id":"n-3255c4dcfe77aa0b","node_b_id":null,"fingerprint":null,"oeis_id":"A375043","in_catalog":0,"target":"reversed_triangle","row_alignment":"node_lengths","rows_matched":7,"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":7,"target":"reversed_triangle","terms_compared":42},"in_catalog":false,"oeis_id":"A375043","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A375043","relation":"match_reversed_triangle"}],"bfiles":[{"node_id":"n-3255c4dcfe77aa0b","oeis_id":"A375043","target":"reversed_triangle","oeis_offset":1,"first_row":1,"rows_used":43,"terms_matched":42,"oeis_terms_agreeing":42,"oeis_terms_available":42,"term_count":989,"raw_size":47394}],"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,           2;\n  2:          2,           6,           4;\n  3:         10,          34,          32,           8;\n  4:        100,         360,         388,         144,         16;\n  5:       1700,        6320,        7316,        3224,        560,        32;\n  6:      44200,      167720,      202856,       98456,      21008,      1952,       64;\n  7:    1635400,     6294040,     7841112,     4048584,     974208,    114240,     6272,     128;\n  8:   81770000,   317972800,   404643680,   218111424,   56807568,   7660416,   542080,   18944,   256;\n  9: 5315050000, 20831772000, 26937784800, 14986529920, 4128714768, 611542176, 50556032, 2315520, 54528, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-2*n+2+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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