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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":"4d2d149b2b06ec97f6de692ad6595ec923df6aeccbdd41281639f3c454529ae7","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-19ce16e7d575fc90","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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,    1;\n  2:  17,   10,     1;\n  3:  43,   51,    15,     1;\n  4:  89,  172,   102,    20,     1;\n  5: 161,  445,   430,   170,    25,     1;\n  6: 265,  966,  1335,   860,   255,    30,    1;\n  7: 407, 1855,  3381,  3115,  1505,   357,   35,   1;\n  8: 593, 3256,  7420,  9016,  6230,  2408,  476,  40,  1;\n  9: 829, 5337, 14652, 22260, 20286, 11214, 3612, 612, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 5 + x, P(2,x) = 17 + 10*x + x^2, P(3,x) = 43 + 51*x + 15*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + x;\nP[2, x_] := 17 + 10*x + x^2;\nP[3, x_] := 43 + 51*x + 15*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 1, 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 + x; v[3] = 17 + 10*x + x^2; v[4] = 43 + 51*x + 15*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*v[i - 4] + (4 + 12*x + 12*x^2 + 4*x^3)*v[i - 3] - (6 + 12*x + 6*x^2)*v[i - 2] + (4 + 4*x)*v[i - 1]); 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-e7d0ffa461d9dbc4","steps":["reverse","D+1"],"edges":["e-68e51a2bf17adc8c","e-29cf85af45210d17"],"text":"A125092 \u2192 reverse \u2192 D+1"},"headline":"A125092 \u2192 reverse \u2192 D+1","chain":null}