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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":"c86c75148a371c97154c69b85eb32e0757288b2adbed6d8879e6ff6fbe744113","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-aff30617856e629d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).","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,      6;\n  3:    66,     60,     12;\n  4:   256,    360,    156,     20;\n  5:   880,   1680,   1140,    320,     30;\n  6:  2784,   6720,   6240,   2760,    570,    42;\n  7:  8288,  24192,  28560,  17360,   5670,   924,    56;\n  8: 23552,  80640, 115584,  89600,  40320, 10416,  1400,   72;\n  9: 64512, 253440, 427392, 403200, 231840, 82656, 17640, 2016, 90;\n","recurrence_sentence":"The row polynomials satisfy (n^3+n-2+(n^3-n^2+4*n-4)*x+(n^2-n)*x^2)*P(n,x) = (2*n^3+6*n^2+8*n+(3*n^3+7*n^2+14*n)*x+(n^3+4*n^2+7*n)*x^2+(n^2+n)*x^3)*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[((2*n^3 + 6*n^2 + 8*n + (3*n^3 + 7*n^2 + 14*n)*x + (n^3 + 4*n^2 + 7*n)*x^2 + (n^2 + n)*x^3)*P[n - 1, x])/(n^3 + n - 2 + (n^3 - n^2 + 4*n - 4)*x + (n^2 - n)*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] = ((2*n^3 + 6*n^2 + 8*n + (3*n^3 + 7*n^2 + 14*n)*x + (n^3 + 4*n^2 + 7*n)*x^2 + (n^2 + n)*x^3)*v[i - 1])/(n^3 + n - 2 + (n^3 - n^2 + 4*n - 4)*x + (n^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-c8c19eba9d68881f","steps":["\u2212D\u00b2+(\u22121+n)D+1+n","x \u21a6 x+1","D"],"edges":["e-168270346f6f7c93","e-2c04e0d67d228006","e-c14e41d48767768e"],"text":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 x \u21a6 x+1 \u2192 D","chain":null}