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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":"16c2afad2a3c7e3b969fe1e0223bf63c06be351eef5caba758463314a4088b1a","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-ff9680a2cfd23e93","role":"sequence","name":"Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.","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:  2,   1;\n  1:  4,   3;\n  2:  6,   9,    2;\n  3:  8,  23,    8;\n  4: 10,  45,   30,    4;\n  5: 12,  75,   92,   20;\n  6: 14, 113,  218,   88,   8;\n  7: 16, 159,  432,  312,  48;\n  8: 18, 213,  758,  860, 240,  16;\n  9: 20, 275, 1220, 1964, 960, 112;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3+2*n^2-(2*n^3+3*n^2+16*n+15)*x-(8*n^2+21*n-20)*x^2-(8*n+32)*x^3)*P(n,x) = -(2*n^3+4*n^2+2*n+(6*n^3-3*n^2-39*n-30)*x-(8*n^3-4*n^2+29*n-25)*x^2-(32*n^2+28*n+44)*x^3-(32*n+32)*x^4)*P(n-1,x) + ((2*n^2+4*n+2)*x+(14*n^2+33*n+15)*x^2-(16*n^2+13)*x^3-(64*n+48)*x^4-64*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 4*n^2 + 2*n + (6*n^3 - 3*n^2 - 39*n - 30)*x - (8*n^3 - 4*n^2 + 29*n - 25)*x^2 - (32*n^2 + 28*n + 44)*x^3 - (32*n + 32)*x^4)*P[n - 1, x] + ((2*n^2 + 4*n + 2)*x + (14*n^2 + 33*n + 15)*x^2 - (16*n^2 + 13)*x^3 - (64*n + 48)*x^4 - 64*x^5)*D[P[n - 1, x], x])/(-2*n^3 - 2*n^2 + (2*n^3 + 3*n^2 + 16*n + 15)*x + (8*n^2 + 21*n - 20)*x^2 + (8*n + 32)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 + 4*n^2 + 2*n + (6*n^3 - 3*n^2 - 39*n - 30)*x - (8*n^3 - 4*n^2 + 29*n - 25)*x^2 - (32*n^2 + 28*n + 44)*x^3 - (32*n + 32)*x^4)*v[i - 1] + ((2*n^2 + 4*n + 2)*x + (14*n^2 + 33*n + 15)*x^2 - (16*n^2 + 13)*x^3 - (64*n + 48)*x^4 - 64*x^5)*deriv(v[i - 1], x))/(-2*n^3 - 2*n^2 + (2*n^3 + 3*n^2 + 16*n + 15)*x + (8*n^2 + 21*n - 20)*x^2 + (8*n + 32)*x^3)); 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-ff9680a2cfd23e93","steps":["D+1"],"edges":["e-f3e51e336b481f24"],"text":"A199478 \u2192 D+1"},"headline":"A199478 \u2192 D+1","chain":null}