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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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"827a4aa832f82242d5e05aef8d0f692091d4d288eb0596472b756ffa82d11ff6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-02156bdfcfec6826","role":"sequence","name":"Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","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,    14,      22;\n  1: 1,    36,     168,       160;\n  2: 1,    82,     834,      2128,       1456;\n  3: 1,   176,    3412,     17344,      29768,       15904;\n  4: 1,   366,   12534,    111728,     360096,      462480,       202672;\n  5: 1,   748,   43104,    626048,    3353216,     7766400,      7946368,       2951680;\n  6: 1,  1514,  142042,   3202832,   26587840,    98551808,    176707072,     150035456,      48361216;\n  7: 1,  3048,  454908,  15389280,  189091848,  1048744608,   2927929344,    4265493504,    3092540160,     880405504;\n  8: 1,  6118, 1428750,  70704048, 1244224272,  9880792272,  40215082224,   89360774400,  109382840064,   69161591296,   17630351872;\n  9: 1, 12260, 4426984, 314358816, 7728766368, 85159095744, 484171320000, 1537565597952, 2825299383168, 2977906000384, 1668889100288, 385148108800;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(2*n+8)*x+(6*n+15)*x^2+(6*n+10)*x^3+(2*n+2)*x^4)*P(n-1,x) + (x+x^2-3*x^3-5*x^4-2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 14*x + 22*x^2.","mathematica":"P[0, x_] := 1 + 14*x + 22*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (2*n + 8)*x + (6*n + 15)*x^2 + (6*n + 10)*x^3 + (2*n + 2)*x^4)*P[n - 1, x] + (x + x^2 - 3*x^3 - 5*x^4 - 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 14*x + 22*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (2*n + 8)*x + (6*n + 15)*x^2 + (6*n + 10)*x^3 + (2*n + 2)*x^4)*v[i - 1] + (x + x^2 - 3*x^3 - 5*x^4 - 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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-02156bdfcfec6826","steps":["x \u21a6 x/(x+1)"],"edges":["e-7d4b92531fc56ec7"],"text":"A156366 \u2192 x \u21a6 x/(x+1)"},"headline":"A156366 \u2192 x \u21a6 x/(x+1)","chain":null}