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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":"bcd947461abef1dbb08441f69212deae2310531133119b025a87d72821d3d4a1","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-f3e0631eb8740f76","role":"sequence","name":"Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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:   0;\n  1:   1;\n  2:   4,     4;\n  3:  10,    32,     18;\n  4:  20,   140,    228,      96;\n  5:  35,   448,   1488,    1712,     600;\n  6:  56,  1176,   6780,   15232,   14040,     4320;\n  7:  84,  2688,  24420,   92032,  160760,   126576,    35280;\n  8: 120,  5544,  74316,  429312, 1227800,  1792128,  1251936,   322560;\n  9: 165, 10560, 199056, 1658384, 7124600, 16717920, 21285936, 13522176, 3265920;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (4*n^2-6*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (6*n^2-12*n+2+(4*n^3-12*n^2+8*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (4*n^2-10*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4 + 4*x, P(3,x) = 10 + 32*x + 18*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 4 + 4*x;\nP[3, x_] := 10 + 32*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 4 + 4*x; v[4] = 10 + 32*x + 18*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 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-f3e0631eb8740f76","steps":["D"],"edges":["e-261f3668d01e679e"],"text":"A358735 \u2192 D"},"headline":"A358735 \u2192 D","chain":null}