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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":"c0c7894ff05d4e49b904e497a074c259206353cfc74bb52d79afeda16aa23d52","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-6b8ac0ed9d98a614","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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;\n  1:           2,            3;\n  2:           8,           26,            21;\n  3:          56,          286,           485,           273;\n  4:         616,         4322,         11341,         13188,           5733;\n  5:        9856,        88248,        315438,        562579,         500556,         177723;\n  6:      216832,      2365264,      10734300,      25940572,       35203129,       25433814,        7642089;\n  7:     6288128,     80952080,     446114748,    1364131688,     2499503345,     2744158959,     1671347979,      435599073;\n  8:   232660736,   3454260304,   22415747516,   83039249060,   192063236989,   283997625668,   262163479230,   138125568168,    31798732329;\n  9: 10702393856, 180068100960, 1345462073400, 5859638480716, 16391480565954, 30541645346727, 37903303980368, 30210652745658, 14032168390422, 2893684641939;\n","recurrence_sentence":"The row polynomials satisfy (2+2*x)*P(n,x) = (n^2+n+2+(3*n^2+3*n+4)*x+(2*n^2+2*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + 2 + (3*n^2 + 3*n + 4)*x + (2*n^2 + 2*n + 2)*x^2)*P[n - 1, x])/(2 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + n + 2 + (3*n^2 + 3*n + 4)*x + (2*n^2 + 2*n + 2)*x^2)*v[i - 1])/(2 + 2*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-ecd77b030b8fab94","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-8bd02eafb0f0a53d","e-3f1b24633323d95d"],"text":"A128813 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A128813 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}