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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":"a6c7653ed6c884dd0ccbb6b858b20124d207fe9e1f4f5a3010f62f77b1dc9b64","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-04fe76d5a46c2282","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","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:      4;\n  2:     17,      1;\n  3:     76,     12;\n  4:    352,     98,      1;\n  5:   1664,    680,     20;\n  6:   7936,   4320,    243,    1;\n  7:  37888,  25984,   2324,   28;\n  8: 180224, 150528,  19264,  452,  1;\n  9: 851968, 847872, 145152, 5520, 36;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-16*n^2+148*n-136+(16*n^2-312*n+296)*x+(16*n-16)*x^2)*P(n,x) = (16*n^3-32*n^2+528*n-512+(n^3+60*n^2-1277*n+1216)*x+(4*n^2-10*n+6)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((32*n^2-32*n-1024)*x-(2*n^2-130*n+2048)*x^2-(8*n-260)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 32*n^2 + 528*n - 512 + (n^3 + 60*n^2 - 1277*n + 1216)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((32*n^2 - 32*n - 1024)*x - (2*n^2 - 130*n + 2048)*x^2 - (8*n - 260)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 16*n^2 + 148*n - 136 + (16*n^2 - 312*n + 296)*x + (16*n - 16)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^3 - 32*n^2 + 528*n - 512 + (n^3 + 60*n^2 - 1277*n + 1216)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((32*n^2 - 32*n - 1024)*x - (2*n^2 - 130*n + 2048)*x^2 - (8*n - 260)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 16*n^2 + 148*n - 136 + (16*n^2 - 312*n + 296)*x + (16*n - 16)*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-3dcaebc50e901d48","steps":["Veronese 2","D+1"],"edges":["e-d1b315cae6550577","e-29e691eaf3c4b4d3"],"text":"A038231 \u2192 Veronese 2 \u2192 D+1"},"headline":"A038231 \u2192 Veronese 2 \u2192 D+1","chain":null}