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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":"92d154651ae7db9b745892f4c85a2718e12272d3b2f156035fb2d1ff62b6c5d1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-24790f2ac2b21c79","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n, k)*binomial(2*n+k, k), 0 <= k <= 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:         26,          40,           15;\n  1:        190,         441,          336,           84;\n  2:       1462,        4536,         5220,         2640,          495;\n  3:      11584,       44990,        69300,        52910,        20020,         3003;\n  4:      93536,      436332,       842205,       860860,       491400,       148512,        18564;\n  5:     765314,     4167765,      9670080,     12390280,      9467640,      4313988,      1085280,       116280;\n  6:    6323270,    39373360,    106714440,    164419920,    157501260,     96047280,     36411144,      7845024,      735471;\n  7:   52638760,   368872650,   1143689040,   2059079400,   2372166720,   1813411908,    919829064,    298514700,    56241900,     4686825;\n  8:  440815036,  3433281600,  11984624190,  24690148560,  33243015960,  30563988840,  19432563150,   8436285000,  2393240850,   400600200,   30045015;\n  9: 3709445084, 31787462938, 123366281280, 286211461800, 441030232500, 473930066610, 362396754720, 197181141300, 74812087350, 18849670125, 2838538560, 193536720;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+14*n+12)*P(n,x) = (22*n^2+71*n+57+(18*n^2+57*n+45)*x)*P(n-1,x) + (17*n+28+(26*n+43)*x+(9*n+15)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 26 + 40*x + 15*x^2.","mathematica":"P[0, x_] := 26 + 40*x + 15*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((22*n^2 + 71*n + 57 + (18*n^2 + 57*n + 45)*x)*P[n - 1, x] + (17*n + 28 + (26*n + 43)*x + (9*n + 15)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 26 + 40*x + 15*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((22*n^2 + 71*n + 57 + (18*n^2 + 57*n + 45)*x)*v[i - 1] + (17*n + 28 + (26*n + 43)*x + (9*n + 15)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12)); 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-24790f2ac2b21c79","steps":["x \u21a6 x+1"],"edges":["e-b36f61198d3f4728"],"text":"A370258 \u2192 x \u21a6 x+1"},"headline":"A370258 \u2192 x \u21a6 x+1","chain":null}