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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":"cf7a84a4c5d6d07dc4e6d427048c128f3102f648935ceb6574fc372acdc68b92","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b2d4a92113cb2b7b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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:      9,       6;\n  3:     25,      18;\n  4:    133,     324,     108;\n  5:    501,    1500,     540;\n  6:   2581,   11970,   14580,    3240;\n  7:  11509,   63126,   94500,   22680;\n  8:  61545,  433608, 1005480,  816480,  136080;\n  9: 308233, 2485944, 6817608, 6804000, 1224720;\n","recurrence_sentence":"The row polynomials satisfy (27*n^2-45*n-15+(108*n-198)*x+108*x^2)*P(n,x) = (54*n^3-117*n^2+33*n-3+(162*n^3-216*n^2-216*n+180)*x+(648*n^2-972*n+432)*x^2+(648*n-648)*x^3)*P(n-1,x) - (36*n^2+12*n+4+(216*n^2+216*n+48)*x+(324*n^2+972*n+324)*x^2+(1296*n+1080)*x^3+1296*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[((54*n^3 - 117*n^2 + 33*n - 3 + (162*n^3 - 216*n^2 - 216*n + 180)*x + (648*n^2 - 972*n + 432)*x^2 + (648*n - 648)*x^3)*P[n - 1, x] - (36*n^2 + 12*n + 4 + (216*n^2 + 216*n + 48)*x + (324*n^2 + 972*n + 324)*x^2 + (1296*n + 1080)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(27*n^2 - 45*n - 15 + (108*n - 198)*x + 108*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] = ((54*n^3 - 117*n^2 + 33*n - 3 + (162*n^3 - 216*n^2 - 216*n + 180)*x + (648*n^2 - 972*n + 432)*x^2 + (648*n - 648)*x^3)*v[i - 1] - (36*n^2 + 12*n + 4 + (216*n^2 + 216*n + 48)*x + (324*n^2 + 972*n + 324)*x^2 + (1296*n + 1080)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(27*n^2 - 45*n - 15 + (108*n - 198)*x + 108*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-c914b69838ac70ec","steps":["basis A013610","D+1"],"edges":["e-3c0217934a464502","e-abcd72f9ca1e1bb3"],"text":"A059344 \u2192 basis A013610 \u2192 D+1"},"headline":"A059344 \u2192 basis A013610 \u2192 D+1","chain":null}