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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":"12559137bc093425b3404fba907e5d11dfa3c1d91b03d91c4f3ba1ba31307db3","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-bbfc5a54ff13a785","role":"sequence","name":"Triangle of coefficients in expansion of (1+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":[],"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:        36,           85,            25;\n  1:       216,         1115,           825,            125;\n  2:      1296,        13345,         18775,           6750,            625;\n  3:      7776,       153275,        367125,         235625,          50000,           3125;\n  4:     46656,      1724905,       6640375,        6783750,        2478125,         346875,          15625;\n  5:    279936,     19207235,     114870525,      175660625,       98350000,       23143750,        2296875,         78125;\n  6:   1679616,    212679265,    1933964575,     4263225750,     3435403125,     1209206250,      198406250,      14687500,        390625;\n  7:  10077696,   2347869995,   32006829525,    99197563625,   110636610000,    54662409375,    13188656250,    1594218750,      91406250,     1953125;\n  8:  60466176,  25876958425,  523848622375,  2243182983750,  3372539678125,  2247717740625,   748103671875,  131306250000,   12175781250,   556640625,    9765625;\n  9: 362797056, 284948873555, 8510962750125, 49726085770625, 98901946550000, 86541948350000, 38170320890625, 9124074609375, 1216617187500, 89267578125, 3330078125, 48828125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 36 + 85*x + 25*x^2.","mathematica":"P[0, x_] := 36 + 85*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[(6 + 5*x)*P[n - 1, x] + 5*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] = 36 + 85*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 5*x)*v[i - 1] + 5*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-bbfc5a54ff13a785","steps":["basis A008277"],"edges":["e-bab24e2df6037a14"],"text":"A013612 \u2192 basis A008277"},"headline":"A013612 \u2192 basis A008277","chain":null}