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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":"d66906f0199679eb7ef17389559a4c9b7e1f79e4fd83ad8e3d3e3dd210e77006","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c1d69b4cd7e1b71e","role":"sequence","name":"basis_map_A013609 of Pascal-(1,3,1) array.","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:      2;\n  2:     14,       8;\n  3:     60,      96,      24;\n  4:    244,     624,     408,      64;\n  5:    910,    3232,    3624,    1408,     160;\n  6:   3282,   14904,   23784,   15936,    4320,     384;\n  7:  11480,   63552,  132240,  128512,   59520,   12288,    896;\n  8:  39368,  257120,  660624,  847744,  572800,  199680,  33152,  2048;\n  9: 132858, 1000128, 3061296, 4900608, 4407360, 2239488, 620928, 86016, 4608;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2+(2*n^2-2*n)*x-(n-1)*x^2)*P(n,x) = -(8*n^2-34*n+22+(14*n^2-74*n+56)*x-(19*n-29)*x^2+4*x^3)*P(n-1,x) + (8*n-8+(32*n-36)*x+(34*n-50)*x^2+(4*n-21)*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 34*n + 22 + (14*n^2 - 74*n + 56)*x - (19*n - 29)*x^2 + 4*x^3)*P[n - 1, x] + (8*n - 8 + (32*n - 36)*x + (34*n - 50)*x^2 + (4*n - 21)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(2*n - 2 + (2*n^2 - 2*n)*x - (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^2 - 34*n + 22 + (14*n^2 - 74*n + 56)*x - (19*n - 29)*x^2 + 4*x^3)*v[i - 1] + (8*n - 8 + (32*n - 36)*x + (34*n - 50)*x^2 + (4*n - 21)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(2*n - 2 + (2*n^2 - 2*n)*x - (n - 1)*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-d8f9ad9ce0e039d0","steps":["basis A013609","D"],"edges":["e-471a582f60d8a81d","e-1e61385793be9b0a"],"text":"A081578 \u2192 basis A013609 \u2192 D"},"headline":"A081578 \u2192 basis A013609 \u2192 D","chain":null}