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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":"c363ee0d20af7dc639d403123468e58dd6ebf62e3d5888a9cf88188d7afa8bd8","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-2bc1923905556bf6","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","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: 256, 185, 25;\n 1: 4096, 5165, 1575, 125;\n 2: 65536, 128945, 66775, 11750, 625;\n 3: 1048576, 3035525, 2380875, 698125, 81250, 3125;\n 4: 16777216, 68988905, 77080375, 33546250, 6415625, 534375, 15625;\n 5: 268435456, 1532653085, 2349034275, 1425335625, 398693750, 53987500, 3390625, 78125;\n 6: 4294967296, 33527892065, 68738156575, 55930575750, 21479653125, 4206956250, 425906250, 20937500, 390625;\n 7: 68719476736, 725560569845, 1954831531275, 2077538631125, 1052920391250, 279883471875, 40626468750, 3197343750, 126562500, 1953125;\n 8: 1099511627776, 15580369350425, 54453422662375, 74177855221250, 48292827240625, 16739824303125, 3268234921875, 366196875000, 23074218750, 751953125, 9765625;\n 9: 17592186044416, 332685314497805, 1493690835973875, 2571780625170625, 2109431056768750, 927796932631250, 234037927921875, 35017215234375, 3123140625000, 161240234375, 4404296875, 48828125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 256 + 185*x + 25*x^2.","mathematica":"P[0, x_] := 256 + 185*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[(16 + 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] = 256 + 185*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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-2bc1923905556bf6","steps":["basis A008277"],"edges":["e-82ec49a96db4a268"],"text":"A038319 \u2192 basis A008277"},"headline":"A038319 \u2192 basis A008277","chain":null}