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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":"2cc22994f30aca35e9694bac2aff7932cea0b90e49179aaa39bec123b505b278","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a9307f854ca152e2","role":"sequence","name":"basis_map_A007318 of Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      6,       3;\n   3:     29,      37,       9;\n   4:    126,     267,     168,       27;\n   5:    516,    1543,    1621,      675,       81;\n   6:   2034,    7884,   11718,     8163,     2538,      243;\n   7:   7807,   37242,   71853,    70903,    36909,     9153,     729;\n   8:  29382,  166593,  395928,   507561,   374502,   155547,   32076,    2187;\n   9: 108923,  715957, 2023677,  3197429,  3066346,  1805823,  623781,  110079,   6561;\n  10: 398988, 2983929, 9781998, 18388935, 21742818, 16647378, 8155134, 2410803, 371790, 19683;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-42*n+39+(39*n^2-126*n+87)*x+(27*n^2-42*n+15)*x^2+(27*n-27)*x^3)*P(n,x) = -(2*n^3-33*n^2+108*n+26-(n^2+20*n-161)*x+(18*n^2-163*n+47)*x^2-(63*n+153)*x^3-81*x^4)*P(n-1,x) + (13*n^2+13*n+(22*n^2+191*n+26)*x+(9*n^2+412*n+213)*x^2+(315*n+421)*x^3+(81*n+315)*x^4+81*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 33*n^2 + 108*n + 26 - (n^2 + 20*n - 161)*x + (18*n^2 - 163*n + 47)*x^2 - (63*n + 153)*x^3 - 81*x^4)*P[n - 1, x] + (13*n^2 + 13*n + (22*n^2 + 191*n + 26)*x + (9*n^2 + 412*n + 213)*x^2 + (315*n + 421)*x^3 + (81*n + 315)*x^4 + 81*x^5)*D[P[n - 1, x], x])/(3*n^3 - 42*n + 39 + (39*n^2 - 126*n + 87)*x + (27*n^2 - 42*n + 15)*x^2 + (27*n - 27)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 33*n^2 + 108*n + 26 - (n^2 + 20*n - 161)*x + (18*n^2 - 163*n + 47)*x^2 - (63*n + 153)*x^3 - 81*x^4)*v[i - 1] + (13*n^2 + 13*n + (22*n^2 + 191*n + 26)*x + (9*n^2 + 412*n + 213)*x^2 + (315*n + 421)*x^3 + (81*n + 315)*x^4 + 81*x^5)*deriv(v[i - 1], x))/(3*n^3 - 42*n + 39 + (39*n^2 - 126*n + 87)*x + (27*n^2 - 42*n + 15)*x^2 + (27*n - 27)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5d39164c4108e043","steps":["basis A007318","D+1"],"edges":["e-0e2d8787d6ee9a58","e-6070df307b92ab13"],"text":"A099097 \u2192 basis A007318 \u2192 D+1"},"headline":"A099097 \u2192 basis A007318 \u2192 D+1","chain":null}