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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":"51fc659425366e4dbc9dde089d92b42f2bbb461a73bf264bf45ef9600566f572","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-c16b107c87dd2b64","role":"sequence","name":"basis_map_A013610 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:      7,       3;\n  2:     37,      42,        9;\n  3:    169,     342,      189,       27;\n  4:    712,    2136,     2079,      756,       81;\n  5:   2860,   11481,    16533,    10476,     2835,      243;\n  6:  11125,   56178,   108513,   101304,    47385,    10206,      729;\n  7:  42289,  257871,   628587,   786726,   539865,   199746,    35721,    2187;\n  8: 157987, 1130280,  3337371,  5276232,  4846230,  2620512,   801171,  122472,   6561;\n  9: 582289, 4784328, 16619985, 31874310, 36974394, 26709831, 11895093, 3096792, 413343, 19683;\n","recurrence_sentence":"The row polynomials satisfy -(81*n^3-639*n^2+741*n+9-(243*n^3-81*n^2+396*n-1692)*x-(243*n^2-405*n-1404)*x^2-(243*n^2+81*n+162)*x^3-(243*n+243)*x^4)*P(n,x) = -(648*n^3-3168*n^2+1626*n+2238+(972*n^3-6804*n^2+720*n+13626)*x-(972*n^2+5832*n-21168)*x^2+(972*n^2-3564*n+3888)*x^3-(486*n+4374)*x^4-1458*x^5)*P(n-1,x) + (351*n^2-312*n-299+(1863*n^2-1188*n-2562)*x+(2673*n^2+459*n-6867)*x^2+(729*n^2+4779*n-4590)*x^3+(3645*n+4293)*x^4+(729*n+3888)*x^5+729*x^6)*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[(-(648*n^3 - 3168*n^2 + 1626*n + 2238 + (972*n^3 - 6804*n^2 + 720*n + 13626)*x - (972*n^2 + 5832*n - 21168)*x^2 + (972*n^2 - 3564*n + 3888)*x^3 - (486*n + 4374)*x^4 - 1458*x^5)*P[n - 1, x] + (351*n^2 - 312*n - 299 + (1863*n^2 - 1188*n - 2562)*x + (2673*n^2 + 459*n - 6867)*x^2 + (729*n^2 + 4779*n - 4590)*x^3 + (3645*n + 4293)*x^4 + (729*n + 3888)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(-81*n^3 + 639*n^2 - 741*n - 9 + (243*n^3 - 81*n^2 + 396*n - 1692)*x + (243*n^2 - 405*n - 1404)*x^2 + (243*n^2 + 81*n + 162)*x^3 + (243*n + 243)*x^4)]];\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] = (-(648*n^3 - 3168*n^2 + 1626*n + 2238 + (972*n^3 - 6804*n^2 + 720*n + 13626)*x - (972*n^2 + 5832*n - 21168)*x^2 + (972*n^2 - 3564*n + 3888)*x^3 - (486*n + 4374)*x^4 - 1458*x^5)*v[i - 1] + (351*n^2 - 312*n - 299 + (1863*n^2 - 1188*n - 2562)*x + (2673*n^2 + 459*n - 6867)*x^2 + (729*n^2 + 4779*n - 4590)*x^3 + (3645*n + 4293)*x^4 + (729*n + 3888)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(-81*n^3 + 639*n^2 - 741*n - 9 + (243*n^3 - 81*n^2 + 396*n - 1692)*x + (243*n^2 - 405*n - 1404)*x^2 + (243*n^2 + 81*n + 162)*x^3 + (243*n + 243)*x^4)); 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-d6b7687ba0ff59dd","steps":["basis A013610","D+1"],"edges":["e-dddeff0317c778fa","e-2358ee2b8fe72927"],"text":"A116412 \u2192 basis A013610 \u2192 D+1"},"headline":"A116412 \u2192 basis A013610 \u2192 D+1","chain":null}