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b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"159c30991c970155e0790d787b08fab6faa55b7358adfd6b0455099104446409","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9a3eef8528def021","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-4ed575ffc61e2c31","kind":"fingerprint_related","node_a_id":"n-25a1bf976309f071","node_b_id":"n-943c738f9048f5dc","fingerprint":"f6-ea33b747599ac881","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ea33b747599ac881"},"other_id":"n-943c738f9048f5dc","other":{"id":"n-943c738f9048f5dc","name":"mobius_x_over_x_plus_one of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","first_index":1,"fingerprint":"f6-ea33b747599ac881","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","8","8"],["1","18","48","32"],["1","38","198","320","160"]],"card_rows":[["1","8","8"],["1","18","48","32"],["1","38","198","320","160"],["1","78","708","2072","2400","960"]],"row_sums_preview":["17","99","717","6219","62877","726219","9433917","136148139"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9a3eef8528def021","steps":["x \u21a6 x/(x+1)"],"edges":["e-d44b22860e00267f"],"text":"A166344 \u2192 x \u21a6 x/(x+1)"},"headline":"A166344 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         8,          8,          1;\n   2:        32,         48,         18,           1;\n   3:       160,        320,        198,          38,          1;\n   4:       960,       2400,       2072,         708,         78,          1;\n   5:      6720,      20160,      22360,       11120,       2358,        158,          1;\n   6:     53760,     188160,     255120,      167400,      53912,       7548,        318,         1;\n   7:    483840,    1935360,    3102960,     2535120,    1106560,     245840,      23598,       638,        1;\n   8:   4838400,   21772800,   40306560,    39466560,   21850080,    6762000,    1077752,     72708,     1278,      1;\n   9:  53222400,  266112000,  558714240,   638184960,  429216480,  171672480,   39198760,   4601840,   221958,   2558,    1;\n  10: 638668800, 3512678400, 8248262400, 10772092800, 8539211520, 4206222720, 1265227440, 219003000, 19295192, 673548, 5118, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+(n+1)*x)*P(n-1,x) - 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