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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":"2918f0e43fb09426837d4936278fd77b1e2563e3fa4fd2d59c28621bcfbeb069","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-00bc3b0d8b47bf75","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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:              5,                3;\n  2:             65,              111,                27;\n  3:           1365,             4971,              3042,               405;\n  4:          39585,           277215,            320373,            101142,              8505;\n  5:        1464645,         18810555,          36638748,          20584827,           3964059,            229635;\n  6:       65909025,       1515556575,        4687649343,        4173293772,        1380937734,         179037297,          7577955;\n  7:     3493178325,     142001656875,      673188770358,      892278265617,      451374443514,       98257821282,       9157327668,       295540245;\n  8:   213083877825,   15209368653375,   108041578883433,   205180034727042,   148933762839639,    48413586844782,    7452677415798,    523202877180,    13299311025;\n  9: 14702787569925, 1835991516174075, 19250787790389168, 51060088232458407, 51123099026300589, 23281624042530957, 5263846960008132, 602970973884378, 33027118095105, 678264862275;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8*n-3+(6*n-3)*x)*P(n-1,x) + (6*n-3)*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[(8*n - 3 + (6*n - 3)*x)*P[n - 1, x] + (6*n - 3)*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (8*n - 3 + (6*n - 3)*x)*v[i - 1] + (6*n - 3)*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2b477f2aa77e3efc","steps":["basis A013610","basis A008277"],"edges":["e-7e513916b57d5183","e-5f42837d6bf999b3"],"text":"A109692 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A109692 \u2192 basis A013610 \u2192 basis A008277","chain":null}