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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":"340de5beecb259ee0408acab7329a4af6448771cb166257dd7016535255684ad","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4ca7e225b5e9bcc6","role":"sequence","name":"Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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: -1,   1;\n  2: -3,   0,   1;\n  3:  0,  -6,   1,    1;\n  4:  5,  -2,  -9,    2,   1;\n  5:  1,  15,  -6,  -12,   3,   1;\n  6: -7,   6,  30,  -12, -15,   4,   1;\n  7: -2, -28,  18,   50, -20, -18,   5,   1;\n  8:  9, -12, -70,   40,  75, -30, -21,   6, 1;\n  9:  3,  45, -40, -140,  75, 105, -42, -24, 7, 1;\n","recurrence_sentence":"The row polynomials satisfy -(10*n^3-24*n^2+38*n-24-(4*n^3-6*n^2-12*n+16)*x-(18*n-12)*x^2+(4*n-2)*x^3)*P(n,x) = (4*n^2-16*n+12-(5*n^3+3*n^2+2*n-8)*x+(2*n^3+2*n^2-2*n-6)*x^2+(7*n+1)*x^3-2*n*x^4)*P(n-1,x) + (20*n^2-12*n+48-(8*n^2+12)*x-(5*n^2-3*n+32)*x^2+(2*n^2+11)*x^3+5*x^4-2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 16*n + 12 - (5*n^3 + 3*n^2 + 2*n - 8)*x + (2*n^3 + 2*n^2 - 2*n - 6)*x^2 + (7*n + 1)*x^3 - 2*n*x^4)*P[n - 1, x] + (20*n^2 - 12*n + 48 - (8*n^2 + 12)*x - (5*n^2 - 3*n + 32)*x^2 + (2*n^2 + 11)*x^3 + 5*x^4 - 2*x^5)*D[P[n - 1, x], x])/(-10*n^3 + 24*n^2 - 38*n + 24 + (4*n^3 - 6*n^2 - 12*n + 16)*x + (18*n - 12)*x^2 - (4*n - 2)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 - 16*n + 12 - (5*n^3 + 3*n^2 + 2*n - 8)*x + (2*n^3 + 2*n^2 - 2*n - 6)*x^2 + (7*n + 1)*x^3 - 2*n*x^4)*v[i - 1] + (20*n^2 - 12*n + 48 - (8*n^2 + 12)*x - (5*n^2 - 3*n + 32)*x^2 + (2*n^2 + 11)*x^3 + 5*x^4 - 2*x^5)*deriv(v[i - 1], x))/(-10*n^3 + 24*n^2 - 38*n + 24 + (4*n^3 - 6*n^2 - 12*n + 16)*x + (18*n - 12)*x^2 - (4*n - 2)*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-4ca7e225b5e9bcc6","steps":["D+1"],"edges":["e-fa558ee223516e16"],"text":"A124038 \u2192 D+1"},"headline":"A124038 \u2192 D+1","chain":null}