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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":"ba3653586f23e4ec796501809a505448bbe46b7e0351f6ec2f8fe05b10877ffc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f4f909be2b0f8048","role":"sequence","name":"basis_map_A013611 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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:          0;\n   2:          8;\n   3:         64,         128;\n   4:        568,        1920,         1536;\n   5:       5952,       26368,        36864,        16384;\n   6:      73032,      385280,       721920,       573440,       163840;\n   7:    1032576,     6177152,     14008320,     15155200,      7864320,      1572864;\n   8:   16569720,   109178496,    284250624,    376750080,    269516800,     99090432,    14680064;\n   9:  297901440,  2121149952,   6142402560,   9436758016,   8349286400,   4271898624,  1174405120,   134217728;\n  10: 5935802760, 45068272128, 142159448064, 244817362944, 253004021760, 161319223296, 62184751104, 13287555072, 1207959552;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(32*n-48)*x+64*x^2)*P(n-2,x) + (4*n-4+16*x)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 8.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (32*n - 48)*x + 64*x^2)*P[n - 2, x] + (4*n - 4 + 16*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 8; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (32*n - 48)*x + 64*x^2)*v[i - 2] + (4*n - 4 + 16*x)*v[i - 1]); 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-7c9c430277778c10","steps":["basis A013611","D"],"edges":["e-1182ea8561b370d4","e-38eabe32192ddf31"],"text":"A161198 \u2192 basis A013611 \u2192 D"},"headline":"A161198 \u2192 basis A013611 \u2192 D","chain":null}