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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":"16bb5427af4b749708b39dca131939055629e0d1b1d03a16558eb5fc18f851d0","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-bf2f08bd0370af8d","role":"sequence","name":"basis_map_A013609 of Pascal-(1,4,1) array.","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:      4,       2;\n  2:     24,      24,       4;\n  3:     96,     184,      80,        8;\n  4:    400,    1120,     880,      224,      16;\n  5:   1536,    5760,    7040,     3264,     576,      32;\n  6:   5824,   27392,   45760,    33024,   10496,    1408,      64;\n  7:  21504,  121984,  260864,   262784,  129536,   30848,    3328,    128;\n  8:  78336,  520704, 1360640,  1777152, 1230336,  451072,   85248,   7680,   256;\n  9: 281600, 2147840, 6645760, 10728448, 9720832, 5004800, 1443840, 225280, 17408, 512;\n","recurrence_sentence":"The row polynomials satisfy (5*n^2+(5*n^3+5*n^2)*x+(18*n^2-10*n)*x^2+(2*n^2+15*n)*x^3+2*n*x^4)*P(n,x) = -(25*n^3-55*n^2-35*n+45+(45*n^3-35*n^2-205*n+145)*x+(142*n^2-221*n+27)*x^2+(18*n^2+91*n-193)*x^3+(14*n-56)*x^4-4*x^5)*P(n-1,x) + (25*n^2+25*n+(100*n^2+190*n+45)*x+(105*n^2+475*n+275)*x^2+(10*n^2+428*n+579)*x^3+(78*n+457)*x^4+(4*n+80)*x^5+4*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[(-(25*n^3 - 55*n^2 - 35*n + 45 + (45*n^3 - 35*n^2 - 205*n + 145)*x + (142*n^2 - 221*n + 27)*x^2 + (18*n^2 + 91*n - 193)*x^3 + (14*n - 56)*x^4 - 4*x^5)*P[n - 1, x] + (25*n^2 + 25*n + (100*n^2 + 190*n + 45)*x + (105*n^2 + 475*n + 275)*x^2 + (10*n^2 + 428*n + 579)*x^3 + (78*n + 457)*x^4 + (4*n + 80)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(5*n^2 + (5*n^3 + 5*n^2)*x + (18*n^2 - 10*n)*x^2 + (2*n^2 + 15*n)*x^3 + 2*n*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] = (-(25*n^3 - 55*n^2 - 35*n + 45 + (45*n^3 - 35*n^2 - 205*n + 145)*x + (142*n^2 - 221*n + 27)*x^2 + (18*n^2 + 91*n - 193)*x^3 + (14*n - 56)*x^4 - 4*x^5)*v[i - 1] + (25*n^2 + 25*n + (100*n^2 + 190*n + 45)*x + (105*n^2 + 475*n + 275)*x^2 + (10*n^2 + 428*n + 579)*x^3 + (78*n + 457)*x^4 + (4*n + 80)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(5*n^2 + (5*n^3 + 5*n^2)*x + (18*n^2 - 10*n)*x^2 + (2*n^2 + 15*n)*x^3 + 2*n*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-ed9434bfd889c409","steps":["basis A013609","D+1"],"edges":["e-175e767cc5e2f6ad","e-f903bab7ae942315"],"text":"A081579 \u2192 basis A013609 \u2192 D+1"},"headline":"A081579 \u2192 basis A013609 \u2192 D+1","chain":null}