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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"d8cab084d6d185e23b6dd7d3e0d3e7b5c25da96f546c28ced7db803bf03c2604","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-8c29b5d8d22f0df0","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.","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:       2,        3;\n  2:      10,       30,         9;\n  3:      40,      180,       198,        27;\n  4:     230,     1380,      2421,      1053,         81;\n  5:    1300,     9750,     24480,     22410,       5022,        243;\n  6:    9040,    81360,    261918,    351108,     167994,      22356,       729;\n  7:   64880,   681240,   2702196,   4938570,    3990384,    1099818,     94770,     2187;\n  8:  536560,  6438720,  30218022,  69142518,   77755788,   37855998,   6539859,   387099,    6561;\n  9: 4641520, 62660520, 340777584, 946422792, 1397274300, 1027882710, 313330032, 36151110, 1535274, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = ((1/2)*n^2+(3/2)*n+1+((3/2)*n^2+(9/2)*n+3)*x)*P(n-2,x) + (2+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[((1/2)*n^2 + (3/2)*n + 1 + ((3/2)*n^2 + (9/2)*n + 3)*x)*P[n - 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