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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":"4743a2217e618ef56a3a49f14f23b163b8adb7dbca4a90678bfe1a327d52715c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cf23198864b5678a","role":"sequence","name":"Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1, 1;\n   3: 1, 3,  1;\n   4: 0, 6,  6,  1;\n   5: 0, 2, 19, 10,   1;\n   6: 0, 0, 15, 45,  15,   1;\n   7: 0, 0,  3, 61,  90,  21,   1;\n   8: 0, 0,  0, 28, 182, 161,  28,   1;\n   9: 0, 0,  0,  4, 141, 448, 266,  36,  1;\n  10: 0, 0,  0,  0,  45, 513, 966, 414, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy -(n^3-5*n^2+8*n-4-(4*n^3-30*n^2+74*n-60)*x-(8*n^2-42*n+52)*x^2-(4*n-8)*x^3)*P(n,x) = (n^3-7*n^2+14*n-8-(7*n^3-45*n^2+92*n-60)*x+(12*n^3-90*n^2+210*n-156)*x^2+(24*n^2-118*n+140)*x^3+(12*n-24)*x^4)*P(n-1,x) - ((2*n^2-6*n+4)*x-(10*n^2-38*n+32)*x^2+(8*n^2-48*n+64)*x^3+(16*n-44)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 7*n^2 + 14*n - 8 - (7*n^3 - 45*n^2 + 92*n - 60)*x + (12*n^3 - 90*n^2 + 210*n - 156)*x^2 + (24*n^2 - 118*n + 140)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] - ((2*n^2 - 6*n + 4)*x - (10*n^2 - 38*n + 32)*x^2 + (8*n^2 - 48*n + 64)*x^3 + (16*n - 44)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-n^3 + 5*n^2 - 8*n + 4 + (4*n^3 - 30*n^2 + 74*n - 60)*x + (8*n^2 - 42*n + 52)*x^2 + (4*n - 8)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = ((n^3 - 7*n^2 + 14*n - 8 - (7*n^3 - 45*n^2 + 92*n - 60)*x + (12*n^3 - 90*n^2 + 210*n - 156)*x^2 + (24*n^2 - 118*n + 140)*x^3 + (12*n - 24)*x^4)*v[i - 1] - ((2*n^2 - 6*n + 4)*x - (10*n^2 - 38*n + 32)*x^2 + (8*n^2 - 48*n + 64)*x^3 + (16*n - 44)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-n^3 + 5*n^2 - 8*n + 4 + (4*n^3 - 30*n^2 + 74*n - 60)*x + (8*n^2 - 42*n + 52)*x^2 + (4*n - 8)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cf23198864b5678a","steps":["D+1"],"edges":["e-feeb8f8915471cc6"],"text":"A098158 \u2192 D+1"},"headline":"A098158 \u2192 D+1","chain":null}