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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":"353fc7e5efeb8ecf221e90870b04e666a8a9a07aa0dbbd4875cfbd36d352d11e","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-d8e527df92e2a449","role":"sequence","name":"basis_map_A013610 of Riordan array (1+2x,x(1+x)).","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:    6,     3;\n  2:   19,    33,      9;\n  3:   46,   165,    144,      27;\n  4:   98,   555,    963,     567,      81;\n  5:  198,  1530,   4158,    4725,    2106,     243;\n  6:  383,  3774,  14103,   24867,   20979,    7533,     729;\n  7:  722,  8649,  41328,  100251,  130086,   87237,   26244,    2187;\n  8: 1333, 18825, 109791,  341145,  607824,  622323,  346275,   89667,   6561;\n  9: 2424, 39435, 271854, 1033101, 2358558, 3306744, 2792070, 1327509, 301806, 19683;\n","recurrence_sentence":"The row polynomials satisfy (27*n^3-63*n^2-21*n+(81*n^2-252*n)*x+(81*n^2-54*n)*x^2+81*n*x^3)*P(n,x) = -(54*n^3-423*n^2+786*n-75-(81*n^2+72*n-1350)*x+(162*n^2-513*n+702)*x^2-(81*n+486)*x^3-243*x^4)*P(n-1,x) + (45*n^2-105*n+10+(162*n^2-243*n-159)*x+(81*n^2+432*n-504)*x^2+(729*n+432)*x^3+(243*n+810)*x^4+243*x^5)*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 - 423*n^2 + 786*n - 75 - (81*n^2 + 72*n - 1350)*x + (162*n^2 - 513*n + 702)*x^2 - (81*n + 486)*x^3 - 243*x^4)*P[n - 1, x] + (45*n^2 - 105*n + 10 + (162*n^2 - 243*n - 159)*x + (81*n^2 + 432*n - 504)*x^2 + (729*n + 432)*x^3 + (243*n + 810)*x^4 + 243*x^5)*D[P[n - 1, x], x])/(27*n^3 - 63*n^2 - 21*n + (81*n^2 - 252*n)*x + (81*n^2 - 54*n)*x^2 + 81*n*x^3)]];\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 - 423*n^2 + 786*n - 75 - (81*n^2 + 72*n - 1350)*x + (162*n^2 - 513*n + 702)*x^2 - (81*n + 486)*x^3 - 243*x^4)*v[i - 1] + (45*n^2 - 105*n + 10 + (162*n^2 - 243*n - 159)*x + (81*n^2 + 432*n - 504)*x^2 + (729*n + 432)*x^3 + (243*n + 810)*x^4 + 243*x^5)*deriv(v[i - 1], x))/(27*n^3 - 63*n^2 - 21*n + (81*n^2 - 252*n)*x + (81*n^2 - 54*n)*x^2 + 81*n*x^3)); 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-c27b11147fce8bfa","steps":["basis A013610","D+1"],"edges":["e-849840c29c946dc7","e-7ad2448b6340d6a4"],"text":"A113214 \u2192 basis A013610 \u2192 D+1"},"headline":"A113214 \u2192 basis A013610 \u2192 D+1","chain":null}