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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"a4803596a8481ac83b189d815d77c16e1f75924770ac90a8b22e86b3c16f0653","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a313e9d783515c70","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+3*x)^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:       1;\n  1:      13,         9;\n  2:      88,       234,        81;\n  3:     496,      2376,      3159,        729;\n  4:    2560,     17856,     42768,      37908,        6561;\n  5:   12544,    115200,    401760,     641520,      426465,       59049;\n  6:   59392,    677376,   3110400,    7231680,     8660520,     4605822,      531441;\n  7:  274432,   3741696,  21337344,   65318400,   113898960,   109122552,    48361131,     4782969;\n  8: 1245184,  19759104, 134701056,  512096256,  1175731200,  1640145024,  1309470624,   497428776,   43046721;\n  9: 5570560, 100859904, 800243712, 3636928512, 10369949184, 19046845440, 22141957824, 15152445792, 5036466357, 387420489;\n","recurrence_sentence":"The row polynomials satisfy (9*n-5+9*x)*P(n,x) = (36*n+16+(81*n+72)*x+81*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[((36*n + 16 + (81*n + 72)*x + 81*x^2)*P[n - 1, x])/(9*n - 5 + 9*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] = ((36*n + 16 + (81*n + 72)*x + 81*x^2)*v[i - 1])/(9*n - 5 + 9*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-6a5af5e1168bb679","steps":["basis A013610","D+1"],"edges":["e-39440d1ff583ab52","e-10587c5b6e93c202"],"text":"A013610 \u2192 basis A013610 \u2192 D+1"},"headline":"A013610 \u2192 basis A013610 \u2192 D+1","chain":null}