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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":"071e400eb49126a46019608e45147c5fc45b54e3ee40d3428c95b3ec8d8dd0b7","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-dd4c9e25ea6d895a","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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:         5,          3;\n  2:        26,         36,          9;\n  3:       180,        348,        189,         27;\n  4:      1584,       3720,       2880,        864,         81;\n  5:     16992,      45552,      43740,      18900,       3645,       243;\n  6:    215424,     637056,     710136,     385560,     107730,     14580,       729;\n  7:   3154176,   10073088,   12554640,    8004528,    2835000,    557928,     56133,     2187;\n  8:  52402176,  178159104,  242439552,  174508992,   73183824,  18370800,   2694384,   209952,   6561;\n  9: 974204928, 3490311168, 5102507520, 4042880640, 1928117520, 576679824, 108387720, 12334680, 767637, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-8*n+4+(12*n-12)*x+9*x^2)*P(n-2,x) + (4*n-2+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n + 4 + (12*n - 12)*x + 9*x^2)*P[n - 2, x] + (4*n - 2 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 8*n + 4 + (12*n - 12)*x + 9*x^2)*v[i - 2] + (4*n - 2 + 6*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013610","D+1"],"edges":["e-45a2b063eb896a2b","e-0c41b14891a0e1d8"],"text":"A028338 \u2192 basis A013610 \u2192 D+1"},"headline":"A028338 \u2192 basis A013610 \u2192 D+1","chain":null}