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h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"c911cdd8fd38fcf988fbd7105ba08cc0ebaeb17e6465704d71f10264db230f5f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-815b996b7a3cb30d","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^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-38f8f9a5c86dbb0b","kind":"fingerprint_related","node_a_id":"n-2fc48ba98a656cdf","node_b_id":"n-e75907fb73d769ed","fingerprint":"f6-b4a8592e069a96fb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b4a8592e069a96fb"},"other_id":"n-e75907fb73d769ed","other":{"id":"n-e75907fb73d769ed","name":"basis_map_A008277 of basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-b4a8592e069a96fb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["441","855","225"],["9261","37395","24300","3375"],["194481","1485135","1800225","587250","50625"]],"card_rows":[["441","855","225"],["9261","37395","24300","3375"],["194481","1485135","1800225","587250","50625"],["4084101","56382075","114088500","65761875","12909375","759375"]],"row_sums_preview":["1521","74331","4117716","253985301","17202656961","1265851691946","100347157885131","8511079483188891"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013610","basis A008277"],"edges":["e-46de8f4df4fd1cb3","e-2bf0d5547860480e"],"text":"A013612 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A013612 \u2192 basis A013610 \u2192 basis A008277","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:              1;\n  1:             28,               20;\n  2:            784,             1520,               400;\n  3:          21952,            88640,             57600,              8000;\n  4:         614656,          4693760,           5689600,           1856000,            160000;\n  5:       17210368,        237593600,         480768000,         277120000,          54400000,           3200000;\n  6:      481890304,      11748700160,       37444096000,       34001920000,       11417600000,        1497600000,         64000000;\n  7:    13492928512,     573575413760,     2781172531200,     3741050880000,     1913139200000,      420044800000,      39424000000,      1280000000;\n  8:   377801998336,   27801478430720,   200591240396800,   384835928064000,   281440051200000,    92028518400000,   14235648000000,   1003520000000,    25600000000;\n  9: 10578455953408, 1342027004641280, 14196233915596800, 37877386477568000, 38092244090880000, 17408451379200000, 3947446272000000, 453304320000000, 24883200000000, 512000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (28+20*x)*P(n-1,x) + 20*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(28 + 20*x)*P[n - 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