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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":"d63535363f04b77499c12a76ab13f2aefbd404fd8a7b12c23deedc568834288f","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-b0d4d18dfd70fd8a","role":"sequence","name":"basis_map_A013610 of Even row Pascal-square 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:      0;\n  1:      6;\n  2:     30,      54;\n  3:    114,     396,     324;\n  4:    390,    1998,    3240,     1620;\n  5:   1260,    8532,   20736,    21060,     7290;\n  6:   3924,   33048,  107244,   165564,   119070,    30618;\n  7:  11910,  119988,  487620,  1014120,  1122660,   612360,   122472;\n  8:  35466,  415962, 2032128,  5324940,  8004420,  6797196,  2939328,   472392;\n  9: 104070, 1392804, 7950960, 25156980, 47907450, 55637280, 37843848, 13384440, 1771470;\n","recurrence_sentence":"The row polynomials satisfy (6*n^2-9*n+3)*P(n,x) = (24*n^2-24*n-3+(54*n^2-63*n-9)*x)*P(n-1,x) - (5*n+27*n*x+36*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((24*n^2 - 24*n - 3 + (54*n^2 - 63*n - 9)*x)*P[n - 1, x] - (5*n + 27*n*x + 36*n*x^2)*D[P[n - 1, x], x])/(6*n^2 - 9*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((24*n^2 - 24*n - 3 + (54*n^2 - 63*n - 9)*x)*v[i - 1] - (5*n + 27*n*x + 36*n*x^2)*deriv(v[i - 1], x))/(6*n^2 - 9*n + 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-79ebafb266da3b41","steps":["basis A013610","D"],"edges":["e-01dc3c5cce6c6f01","e-ae40ee5fb0cfc7a6"],"text":"A172431 \u2192 basis A013610 \u2192 D"},"headline":"A172431 \u2192 basis A013610 \u2192 D","chain":null}