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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":"cfcb5881be4fa491e06da573dad6fbe222693d066daab351c10c10abe5c5b53c","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-6bdb5d7e28f56300","role":"sequence","name":"basis_map_A007318 of Triangular array formed by the Mersenne numbers.","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,      1;\n  2:     22,     10,      1;\n  3:     92,     68,     15,      1;\n  4:    372,    388,    138,     20,      1;\n  5:   1468,   1996,   1000,    232,     25,     1;\n  6:   5688,   9584,   6264,   2040,    350,    30,    1;\n  7:  21728,  43808,  35544,  15080,   3620,   492,   35,   1;\n  8:  82064, 192992, 187760,  98864,  30860,  5852,  658,  40,  1;\n  9: 307088, 826192, 939552, 593440, 229684, 56532, 8848, 848, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+6*n-10*n*x-2*n*x^2)*P(n,x) = (4*n^3+12*n^2+16*n+8+(n^3+3*n^2-22*n-24)*x-(10*n+10)*x^2-(n+1)*x^3)*P(n-1,x) + (8*n^2+8*n-32+(8*n^2+8*n-88)*x+(n^2+n-68)*x^2-15*x^3-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[((4*n^3 + 12*n^2 + 16*n + 8 + (n^3 + 3*n^2 - 22*n - 24)*x - (10*n + 10)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (8*n^2 + 8*n - 32 + (8*n^2 + 8*n - 88)*x + (n^2 + n - 68)*x^2 - 15*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 6*n - 10*n*x - 2*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] = ((4*n^3 + 12*n^2 + 16*n + 8 + (n^3 + 3*n^2 - 22*n - 24)*x - (10*n + 10)*x^2 - (n + 1)*x^3)*v[i - 1] + (8*n^2 + 8*n - 32 + (8*n^2 + 8*n - 88)*x + (n^2 + n - 68)*x^2 - 15*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 6*n - 10*n*x - 2*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-9b5c7e97e5191da6","steps":["basis A007318","D+1"],"edges":["e-1d7f8f614531604f","e-f535dfcd61d51a51"],"text":"A110441 \u2192 basis A007318 \u2192 D+1"},"headline":"A110441 \u2192 basis A007318 \u2192 D+1","chain":null}
