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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"1fdcaefb241a440f5e2391ac8de41de14f4d4cb204e7d28949e9b8100da81d11","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-47b361d1e904fbeb","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (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-030988d9b53a393e","kind":"fingerprint_related","node_a_id":"n-3ac8a6274ff270bb","node_b_id":"n-91aecd3506520caf","fingerprint":"f6-9247102c6b6e4eea","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9247102c6b6e4eea"},"other_id":"n-3ac8a6274ff270bb","other":{"id":"n-3ac8a6274ff270bb","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-9247102c6b6e4eea","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","5"],["16","44","29"]],"card_rows":[["1"],["4","5"],["16","44","29"],["64","272","376","169"]],"row_sums_preview":["1","9","89","881","8721","86329","854569","8459361"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-67136cef6b741eb8","e-2e912dcfc372c834"],"text":"A103327 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A103327 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:        29,         44,         16;\n  1:       169,        376,        272,          64;\n  2:       985,       2888,       3120,        1472,         256;\n  3:      5741,      20892,      30000,       21248,        7424,        1024;\n  4:     33461,     145428,     260448,      246016,      129280,       35840,        4096;\n  5:    195025,     985520,    2114400,     2496640,     1752320,      731136,      167936,      16384;\n  6:   1136689,    6547792,   16368032,    23191424,    20371200,    11360256,     3928064,     770048,      65536;\n  7:   6625109,   42847988,  122284960,   202124032,   213240576,   148915200,    68841472,   20316160,    3473408,    262144;\n  8:  38613965,  277040572,  888733680,  1678692608,  2067568384,  1735093248,  1004781568,  396492800,  102039552,  15466496,  1048576;\n  9: 225058681, 1773851304, 6318279408, 13424966336, 18906940160, 18531917824, 12900220928, 6377766912, 2194735104, 500695040, 68157440, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 29 + 44*x + 16*x^2, P(1,x) = 169 + 376*x + 272*x^2 + 64*x^3.","mathematica":"P[0, x_] := 29 + 44*x + 16*x^2;\nP[1, x_] := 169 + 376*x + 272*x^2 + 64*x^3;\nP[n_, x_] := P[n, x] = Expand[-P[n - 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