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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":"2fe95954792f9c33f14a6bd0d94f91adcfee506b897b2ca0373a127fdecfd2ae","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-6a690be6f27c7004","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.","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,  1;\n  1:  2,  1;\n  2:  1,  5,   5,   1;\n  3:  4,  9,   6,   1;\n  4:  1,  9,  22,  22,   9,   1;\n  5:  6, 25,  40,  30,  10,   1;\n  6:  1, 13,  51,  95,  95,  51,  13,   1;\n  7:  8, 49, 126, 175, 140,  63,  14,   1;\n  8:  1, 17,  92, 252, 406, 406, 252,  92, 17, 1;\n  9: 10, 81, 288, 588, 756, 630, 336, 108, 18, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n-1+2*n*x+x^2)*P(n,x) = -(n^3-n^2-3*n+2-(n^2+5*n-1)*x-(n^3+2*n^2+3*n+2)*x^2-(2*n^2+2*n+1)*x^3-n*x^4)*P(n-1,x) + (n^2+n-1+(n^2+n-3)*x-(n^2+3*n+5)*x^2-(n^2+5*n+6)*x^3-(2*n+4)*x^4-x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + x.","mathematica":"P[0, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 3*n + 2 - (n^2 + 5*n - 1)*x - (n^3 + 2*n^2 + 3*n + 2)*x^2 - (2*n^2 + 2*n + 1)*x^3 - n*x^4)*P[n - 1, x] + (n^2 + n - 1 + (n^2 + n - 3)*x - (n^2 + 3*n + 5)*x^2 - (n^2 + 5*n + 6)*x^3 - (2*n + 4)*x^4 - x^5)*D[P[n - 1, x], x])/(n^2 + n - 1 + 2*n*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - 3*n + 2 - (n^2 + 5*n - 1)*x - (n^3 + 2*n^2 + 3*n + 2)*x^2 - (2*n^2 + 2*n + 1)*x^3 - n*x^4)*v[i - 1] + (n^2 + n - 1 + (n^2 + n - 3)*x - (n^2 + 3*n + 5)*x^2 - (n^2 + 5*n + 6)*x^3 - (2*n + 4)*x^4 - x^5)*deriv(v[i - 1], x))/(n^2 + n - 1 + 2*n*x + 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-6a690be6f27c7004","steps":["D+1"],"edges":["e-becd80478184fd2d"],"text":"Veronese parent of A109446 \u2192 D+1"},"headline":"Veronese parent of A109446 \u2192 D+1","chain":null}