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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e6fa3916511b6918e73cb63b1ec7266036786bad28e24d7dc9fb24857db26982","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-4d60c3dd8627c01d","role":"sequence","name":"Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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-521e9d8c39283dbd","kind":"oeis_match","node_a_id":"n-e63fe40b997d2bc1","node_b_id":null,"fingerprint":null,"oeis_id":"A008548","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A008548","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A008548","relation":"match_row_sums"}],"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:       6,        28,         32;\n  3:      24,       184,        464,        384;\n  4:     120,      1304,       5264,       9344,        6144;\n  5:     720,     10224,      57664,     161344,      223744,       122880;\n  6:    5040,     88848,     649024,    2513344,     5438464,      6230016,      2949120;\n  7:   40320,    851904,    7679936,   38279424,   113881344,    202117120,    198033408,     82575360;\n  8:  362880,   8957376,   96380352,  590272768,  2249873664,   5463257088,   8250048512,   7080247296,   2642411520;\n  9: 3628800, 102637440, 1286269056, 9372420352, 43748556288, 135628022784, 279177740288, 367804219392, 281313017856, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+4*n*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 + 1 + 4*n*x)*P[n - 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