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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":"e02fa80907bea9b2d31498a2c3f55e62128151af1d57b42069d17792c1806d04","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-a9d89adf599d215a","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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:            1;\n  1:           19,            35;\n  2:          361,          1330,           1225;\n  3:         6859,         37905,          69825,           42875;\n  4:       130321,        960260,        2653350,         3258500,         1500625;\n  5:      2476099,      22806175,       84022750,       154778750,       142559375,        52521875;\n  6:     47045881,     519980790,     2394648375,      5881592500,      8125884375,      5987493750,       1838265625;\n  7:    893871739,   11526240845,    63697646775,    195562950625,    360247540625,    398168334375,     244489328125,     64339296875;\n  8:  16983563041,  250284086920,  1613673718300,   5945113699000,  13689406543750,  20173862275000,   18581188937500,   9779573125000,   2251875390625;\n  9: 322687697779, 5349822357915, 39419743689900, 169435740421500, 468177703796250, 862432612256250, 1059127769437500, 836153502187500, 385070691796875, 78815638671875;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (19+54*x+35*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[((19 + 54*x + 35*x^2)*P[n - 1, x])/(1 + 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] = ((19 + 54*x + 35*x^2)*v[i - 1])/(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-2f02b2035f8f4541","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-6db2596ebcec1df1","e-dd3de843fbc19c0b"],"text":"A038322 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038322 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}