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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":"7698d592da35ad9aeb28078e755ac2c8061736a217c36d4a53fa7cc0f3dcef46","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-f339d17ab0e2d2a9","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","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,        2;\n  2:      23,       21,        4;\n  3:     108,      157,       67,        8;\n  4:     534,     1075,      733,      193,       16;\n  5:    2803,     7261,     6913,     2933,      531,       32;\n  6:   15623,    49742,    61423,    36788,    10881,     1433,      64;\n  7:   92249,   349728,   534245,   418778,   177166,    38841,    3851,    128;\n  8:  575341,  2537297,  4639171,  4530277,  2533296,   806153,  136109,  10401,   256;\n  9: 3778708, 19041277, 40667833, 47792573, 33541154, 14257619, 3549631, 473677, 28387, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+2+2*x)*P(n,x) = -(n^3+5*n^2+6*n+(2*n^3+11*n^2+13*n)*x+(n^3+9*n^2+13*n+3)*x^2+(3*n^2+7*n+2)*x^3+2*n*x^4)*P(n-2,x) + (n^2+5*n+6+(3*n^2+17*n+22)*x+(3*n^2+22*n+33)*x^2+(n^2+13*n+26)*x^3+(3*n+11)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2+7*n+6+(2*n^2+10*n+8)*x+(5*n+5)*x^2+2*x^3)*P(n-1,x) - (n+2+(2*n+6)*x+(n+6)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 5*n^2 + 6*n + (2*n^3 + 11*n^2 + 13*n)*x + (n^3 + 9*n^2 + 13*n + 3)*x^2 + (3*n^2 + 7*n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] + (n^2 + 5*n + 6 + (3*n^2 + 17*n + 22)*x + (3*n^2 + 22*n + 33)*x^2 + (n^2 + 13*n + 26)*x^3 + (3*n + 11)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 + 7*n + 6 + (2*n^2 + 10*n + 8)*x + (5*n + 5)*x^2 + 2*x^3)*P[n - 1, x] - (n + 2 + (2*n + 6)*x + (n + 6)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 5*n^2 + 6*n + (2*n^3 + 11*n^2 + 13*n)*x + (n^3 + 9*n^2 + 13*n + 3)*x^2 + (3*n^2 + 7*n + 2)*x^3 + 2*n*x^4)*v[i - 2] + (n^2 + 5*n + 6 + (3*n^2 + 17*n + 22)*x + (3*n^2 + 22*n + 33)*x^2 + (n^2 + 13*n + 26)*x^3 + (3*n + 11)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 + 7*n + 6 + (2*n^2 + 10*n + 8)*x + (5*n + 5)*x^2 + 2*x^3)*v[i - 1] - (n + 2 + (2*n + 6)*x + (n + 6)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2 + 2*x)); 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-0990b5be9f344750","steps":["x \u21a6 x+1","D+1"],"edges":["e-2b15edbdb8f7710e","e-7a66bd1f2e9eca5c"],"text":"A126351 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A126351 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}