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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":"902819e54ad4c624cecc878b512774c3698970939747201448cf606311fde980","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-37"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-44b1dbf5d009bf35","role":"sequence","name":"A convolution triangle of numbers obtained from A036224.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-36"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-36, 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:           379,            44,             1;\n  1:          5713,          1242,            66,             1;\n  2:         80032,         28102,          2589,            88,            1;\n  3:       1080325,        555077,         77815,          4420,          110,           1;\n  4:      14354557,      10012512,       1975443,        165500,         6735,         132,          1;\n  5:     189853438,     169546039,      44572143,       5125694,       301805,        9534,        154,        1;\n  6:    2511177679,    2744639366,     922030813,     138677168,     11024650,      497378,      12817,      176,       1;\n  7:   33258000301,   43005575310,   17852826033,    3388726806,    347541537,    20925387,     762867,    16584,     198,     1;\n  8:  440953665088,  657853033564,  328323375558,   76477957680,   9790928625,   752828832,   36315237,  1108920,   20835,   220,   1;\n  9: 5849978617585, 9882747323963, 5796735000301, 1619545476081, 252345070860, 24011602581, 1467805899, 58915788, 1546185, 25570, 242, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+3)*P(n,x) = (6*n+50+2*x)*P(n-1,x) + (37+38*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 379 + 44*x + x^2.","mathematica":"P[0, x_] := 379 + 44*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 50 + 2*x)*P[n - 1, x] + (37 + 38*x + x^2)*D[P[n - 1, x], x])/(n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 379 + 44*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 50 + 2*x)*v[i - 1] + (37 + 38*x + x^2)*deriv(v[i - 1], x))/(n + 3)); 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-44b1dbf5d009bf35","steps":["x \u21a6 x+1"],"edges":["e-62b3ce2dd6097757"],"text":"A132166 \u2192 x \u21a6 x+1"},"headline":"A132166 \u2192 x \u21a6 x+1","chain":null}