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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":"3d823aefa42a2e18b4aed0a558c60011a99d411cb0fca8c78af2cdf6791ff84e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f22372ef3b33093f","role":"sequence","name":"basis_map_A013610 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:       8,        3;\n  2:      52,       48,        9;\n  3:     302,      477,      216,        27;\n  4:    1639,     3768,     2889,       864,        81;\n  5:    8516,    26016,    28908,     14526,      3240,      243;\n  6:   42943,   164592,   242811,    176040,     65610,    11664,      729;\n  7:  211886,   979851,  1814940,   1743255,    933930,   276291,    40824,    2187;\n  8: 1028356,  5579040, 12486645,  15047640,  10656360,  4517856,  1107351,  139968,   6561;\n  9: 4926740, 30705165, 80761860, 117609570, 104373360, 58373217, 20452824, 4277772, 472392, 19683;\n","recurrence_sentence":"The row polynomials satisfy (54*n^3+138*n-18*n*x-54*n*x^2)*P(n,x) = (135*n^3+405*n^2+633*n+363+(81*n^3+243*n^2+135*n-27)*x-(243*n+243)*x^2-(81*n+81)*x^3)*P(n-1,x) + (117*n^2+117*n+26+(270*n^2+270*n-213)*x+(81*n^2+81*n-729)*x^2-459*x^3-81*x^4)*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[((135*n^3 + 405*n^2 + 633*n + 363 + (81*n^3 + 243*n^2 + 135*n - 27)*x - (243*n + 243)*x^2 - (81*n + 81)*x^3)*P[n - 1, x] + (117*n^2 + 117*n + 26 + (270*n^2 + 270*n - 213)*x + (81*n^2 + 81*n - 729)*x^2 - 459*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 + 138*n - 18*n*x - 54*n*x^2)]];\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] = ((135*n^3 + 405*n^2 + 633*n + 363 + (81*n^3 + 243*n^2 + 135*n - 27)*x - (243*n + 243)*x^2 - (81*n + 81)*x^3)*v[i - 1] + (117*n^2 + 117*n + 26 + (270*n^2 + 270*n - 213)*x + (81*n^2 + 81*n - 729)*x^2 - 459*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 + 138*n - 18*n*x - 54*n*x^2)); 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-6466a8550c08a52f","steps":["basis A013610","D+1"],"edges":["e-5356faecb6c80906","e-ca0253591ee9bc13"],"text":"A116414 \u2192 basis A013610 \u2192 D+1"},"headline":"A116414 \u2192 basis A013610 \u2192 D+1","chain":null}