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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":"58bea866e5ba6767ad0896fc03c96aff3626af0a6a5e1433a2a849d8cde9649f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ee90405a2d071e0a","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:     1;\n  2:     5,      3;\n  3:    20,     33,      9;\n  4:    70,    201,    153,      27;\n  5:   229,    951,   1269,     621,      81;\n  6:   719,   3924,   7641,    6534,    2349,     243;\n  7:  2195,  14868,  38367,   47898,   29970,    8505,    729;\n  8:  6565,  53151, 171423,  284175,  259200,  127575,  29889,   2187;\n  9: 19330, 182145, 706149, 1466721, 1776330, 1272105, 515403, 102789, 6561;\n","recurrence_sentence":"The row polynomials satisfy (54*n^3-243*n^2+393*n-225-(54*n-81)*x^2)*P(n,x) = (81*n^3-189*n^2+126*n-39+(81*n^3-162*n^2-18*n+99)*x-(162*n-189)*x^2-(81*n-81)*x^3)*P(n-1,x) + (45*n^2-90*n-10+(162*n^2-324*n-126)*x+(81*n^2-162*n-387)*x^2-324*x^3-81*x^4)*d/dx 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[((81*n^3 - 189*n^2 + 126*n - 39 + (81*n^3 - 162*n^2 - 18*n + 99)*x - (162*n - 189)*x^2 - (81*n - 81)*x^3)*P[n - 1, x] + (45*n^2 - 90*n - 10 + (162*n^2 - 324*n - 126)*x + (81*n^2 - 162*n - 387)*x^2 - 324*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 - 243*n^2 + 393*n - 225 - (54*n - 81)*x^2)]];\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] = ((81*n^3 - 189*n^2 + 126*n - 39 + (81*n^3 - 162*n^2 - 18*n + 99)*x - (162*n - 189)*x^2 - (81*n - 81)*x^3)*v[i - 1] + (45*n^2 - 90*n - 10 + (162*n^2 - 324*n - 126)*x + (81*n^2 - 162*n - 387)*x^2 - 324*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 - 243*n^2 + 393*n - 225 - (54*n - 81)*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-0eba401b117f16a9","steps":["basis A013610","D+1"],"edges":["e-0b8f676aef1665e7","e-e89ee81e29043811"],"text":"A085478 \u2192 basis A013610 \u2192 D+1"},"headline":"A085478 \u2192 basis A013610 \u2192 D+1","chain":null}