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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":"b0222ee5481e1249120524d563a7b01528f5441371e8078df6073db8de66d3fc","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-5fccad4e67ef9278","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^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:      3;\n  2:     13,       4;\n  3:     63,      36;\n  4:    297,     248,     16;\n  5:   1323,    1560,    240;\n  6:   5589,    9180,   2352,     64;\n  7:  22599,   50652,  19152,   1344;\n  8:  88209,  263088, 139104,  17152,  256;\n  9: 334611, 1294704, 925344, 172800, 6912;\n","recurrence_sentence":"The row polynomials satisfy (6*n^3-24*n^2+57*n-39+(24*n^2-138*n+114)*x+(24*n-24)*x^2)*P(n,x) = (18*n^3-36*n^2+99*n-81+(8*n^3+40*n^2-426*n+378)*x+(32*n^2-80*n+48)*x^2+(32*n-32)*x^3)*P(n-1,x) + ((36*n^2-36*n-162)*x-(16*n^2-160*n+324)*x^2-(64*n-320)*x^3-64*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n^3 - 36*n^2 + 99*n - 81 + (8*n^3 + 40*n^2 - 426*n + 378)*x + (32*n^2 - 80*n + 48)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] + ((36*n^2 - 36*n - 162)*x - (16*n^2 - 160*n + 324)*x^2 - (64*n - 320)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(6*n^3 - 24*n^2 + 57*n - 39 + (24*n^2 - 138*n + 114)*x + (24*n - 24)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((18*n^3 - 36*n^2 + 99*n - 81 + (8*n^3 + 40*n^2 - 426*n + 378)*x + (32*n^2 - 80*n + 48)*x^2 + (32*n - 32)*x^3)*v[i - 1] + ((36*n^2 - 36*n - 162)*x - (16*n^2 - 160*n + 324)*x^2 - (64*n - 320)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(6*n^3 - 24*n^2 + 57*n - 39 + (24*n^2 - 138*n + 114)*x + (24*n - 24)*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-01cea33b6485d9cc","steps":["Veronese 2","D+1"],"edges":["e-4fd32b1a93a66d21","e-212f4c4cdad7ef6e"],"text":"A038220 \u2192 Veronese 2 \u2192 D+1"},"headline":"A038220 \u2192 Veronese 2 \u2192 D+1","chain":null}