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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":"93ffaf9669812e28da98fa8e8e16c5eb67c968bcde106297e0c1ae4a8dbeb65e","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-37ca9f2607503703","role":"sequence","name":"basis_map_A013609 of Pascal-(1,2,1) array read by antidiagonals.","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:    18,     20,       4;\n  3:    64,    128,      64,       8;\n  4:   220,    640,     568,     176,      16;\n  5:   720,   2776,    3680,    2016,     448,      32;\n  6:  2296,  11056,   19712,   16192,    6304,    1088,     64;\n  7:  7168,  41472,   93440,  104128,   60672,   18176,   2560,    128;\n  8: 22032, 148992,  406848,  577152,  457536,  204288,  49536,   5888,   256;\n  9: 66880, 517920, 1664256, 2879232, 2917632, 1772160, 637440, 129536, 13312, 512;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2+(3*n^3+3*n^2)*x+(10*n^2-6*n)*x^2+(2*n^2+7*n)*x^3+2*n*x^4)*P(n,x) = -(9*n^3-21*n^2-15*n+15+(15*n^3-21*n^2-75*n+51)*x+(38*n^2-83*n+17)*x^2+(10*n^2+11*n-65)*x^3+(6*n-32)*x^4-4*x^5)*P(n-1,x) + (9*n^2+9*n+(36*n^2+66*n+15)*x+(39*n^2+165*n+93)*x^2+(6*n^2+160*n+203)*x^3+(46*n+177)*x^4+(4*n+48)*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[(-(9*n^3 - 21*n^2 - 15*n + 15 + (15*n^3 - 21*n^2 - 75*n + 51)*x + (38*n^2 - 83*n + 17)*x^2 + (10*n^2 + 11*n - 65)*x^3 + (6*n - 32)*x^4 - 4*x^5)*P[n - 1, x] + (9*n^2 + 9*n + (36*n^2 + 66*n + 15)*x + (39*n^2 + 165*n + 93)*x^2 + (6*n^2 + 160*n + 203)*x^3 + (46*n + 177)*x^4 + (4*n + 48)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(3*n^2 + (3*n^3 + 3*n^2)*x + (10*n^2 - 6*n)*x^2 + (2*n^2 + 7*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] = (-(9*n^3 - 21*n^2 - 15*n + 15 + (15*n^3 - 21*n^2 - 75*n + 51)*x + (38*n^2 - 83*n + 17)*x^2 + (10*n^2 + 11*n - 65)*x^3 + (6*n - 32)*x^4 - 4*x^5)*v[i - 1] + (9*n^2 + 9*n + (36*n^2 + 66*n + 15)*x + (39*n^2 + 165*n + 93)*x^2 + (6*n^2 + 160*n + 203)*x^3 + (46*n + 177)*x^4 + (4*n + 48)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(3*n^2 + (3*n^3 + 3*n^2)*x + (10*n^2 - 6*n)*x^2 + (2*n^2 + 7*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-f25cd955a876cde3","steps":["basis A013609","D+1"],"edges":["e-144ed6eafdc58374","e-1c060ea1c35b0b78"],"text":"A081577 \u2192 basis A013609 \u2192 D+1"},"headline":"A081577 \u2192 basis A013609 \u2192 D+1","chain":null}