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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"19e3a73477051725b5fcb5f98f7a75b5004cb673d2ca50ec1984429d20321d28","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-a3274460eb3b9452","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*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:         18,          54,           9;\n  3:         96,         432,         450,           27;\n  4:       1182,        7092,       11601,         2889,           81;\n  5:      11100,       83250,      203364,       165888,        16200,          243;\n  6:     187680,     1689120,     5259258,      6218748,      1754784,        83106,         729;\n  7:    2639760,    27717480,   108021132,    186515190,    126994068,     15394536,      399492,      2187;\n  8:   58768320,   705219840,  3242280834,   6966102546,   6630676686,   1935796806,   118245987,   1826145,    6561;\n  9: 1133844240, 15306897240, 82202046408, 220231803204, 298477421640, 176368668570, 23978375892, 820645506, 8017542, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = ((1/3)*n^3+(3/2)*n^2+(13/6)*n+1+(n^3+(9/2)*n^2+(13/2)*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[((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1 + (n^3 + (9/2)*n^2 + (13/2)*n + 3)*x)*P[n - 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