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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":"530e83493d20216461e6b59aebbf44a14ae8d0c74ec0a2b03eafdb6e60405440","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-6aadffa116a44288","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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:       9,        6;\n  2:      54,       96,        27;\n  3:     288,      906,       657,       108;\n  4:    1440,     6576,      8820,      3672,       405;\n  5:    6912,    41568,     83808,     65772,     18225,      1458;\n  6:   32256,   241152,    655488,    790992,    416826,     83592,      5103;\n  7:  147456,  1319424,   4531968,   7502976,   6174144,   2364390,    362313,    17496;\n  8:  663552,  6918144,  28781568,  61005312,  69802560,  42200352,  12369672,  1504656,   59049;\n  9: 2949120, 35119104, 171786240, 445436928, 659508480, 560260800, 261320256, 60833592, 6042681, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-15*n^2+9*n-(6*n^3-17*n^2+30*n-19)*x-(12*n^2-19*n+6)*x^2-6*n*x^3)*P(n,x) = -(24*n^3-36*n^2-24*n+36+(30*n^3-49*n^2-125*n+144)*x-(54*n^3-87*n^2+98*n-74)*x^2-(108*n^2-129*n+69)*x^3-(54*n-18)*x^4)*P(n-1,x) + ((12*n^2-12)*x+(24*n^2+16*n-46)*x^2-(36*n^2-24*n+16)*x^3-(72*n-30)*x^4-36*x^5)*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[(-(24*n^3 - 36*n^2 - 24*n + 36 + (30*n^3 - 49*n^2 - 125*n + 144)*x - (54*n^3 - 87*n^2 + 98*n - 74)*x^2 - (108*n^2 - 129*n + 69)*x^3 - (54*n - 18)*x^4)*P[n - 1, x] + ((12*n^2 - 12)*x + (24*n^2 + 16*n - 46)*x^2 - (36*n^2 - 24*n + 16)*x^3 - (72*n - 30)*x^4 - 36*x^5)*D[P[n - 1, x], x])/(-6*n^3 + 15*n^2 - 9*n + (6*n^3 - 17*n^2 + 30*n - 19)*x + (12*n^2 - 19*n + 6)*x^2 + 6*n*x^3)]];\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] = (-(24*n^3 - 36*n^2 - 24*n + 36 + (30*n^3 - 49*n^2 - 125*n + 144)*x - (54*n^3 - 87*n^2 + 98*n - 74)*x^2 - (108*n^2 - 129*n + 69)*x^3 - (54*n - 18)*x^4)*v[i - 1] + ((12*n^2 - 12)*x + (24*n^2 + 16*n - 46)*x^2 - (36*n^2 - 24*n + 16)*x^3 - (72*n - 30)*x^4 - 36*x^5)*deriv(v[i - 1], x))/(-6*n^3 + 15*n^2 - 9*n + (6*n^3 - 17*n^2 + 30*n - 19)*x + (12*n^2 - 19*n + 6)*x^2 + 6*n*x^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-c85abc56078671cc","steps":["basis A013610","D+1"],"edges":["e-12f73d6f0ea24cf5","e-db10e7d20bfef97f"],"text":"A232535 \u2192 basis A013610 \u2192 D+1"},"headline":"A232535 \u2192 basis A013610 \u2192 D+1","chain":null}