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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":"266a2654af3a0349df81d9cf61f720d48ccd3cb2cebaa275f0a8afb26557d38f","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-6fa6977267f2e73d","role":"sequence","name":"Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).","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: 1,  1;\n  2: 2,  4,  1;\n  3: 0, 12,  9,   1;\n  4: 0, 12, 42,  16,    1;\n  5: 0,  0, 90, 110,   25,     1;\n  6: 0,  0, 60, 380,  240,    36,    1;\n  7: 0,  0,  0, 560, 1190,   462,   49,    1;\n  8: 0,  0,  0, 280, 2870,  3080,  812,   64,  1;\n  9: 0,  0,  0,   0, 3150, 10710, 6972, 1332, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy -(n^3-n^2-(n^3-6*n^2+7*n)*x-(2*n^2-7*n)*x^2-n*x^3)*P(n,x) = (4*n^3-16*n^2+12*n-(7*n^3-33*n^2+48*n-24)*x+(3*n^3-26*n^2+52*n-32)*x^2+(6*n^2-26*n+16)*x^3+(3*n-2)*x^4)*P(n-1,x) - ((8*n^2-8*n)*x-(10*n^2-38*n+16)*x^2+(2*n^2-25*n+44)*x^3+(4*n-18)*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^3 - 16*n^2 + 12*n - (7*n^3 - 33*n^2 + 48*n - 24)*x + (3*n^3 - 26*n^2 + 52*n - 32)*x^2 + (6*n^2 - 26*n + 16)*x^3 + (3*n - 2)*x^4)*P[n - 1, x] - ((8*n^2 - 8*n)*x - (10*n^2 - 38*n + 16)*x^2 + (2*n^2 - 25*n + 44)*x^3 + (4*n - 18)*x^4 + 2*x^5)*D[P[n - 1, x], x])/(-n^3 + n^2 + (n^3 - 6*n^2 + 7*n)*x + (2*n^2 - 7*n)*x^2 + n*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^3 - 16*n^2 + 12*n - (7*n^3 - 33*n^2 + 48*n - 24)*x + (3*n^3 - 26*n^2 + 52*n - 32)*x^2 + (6*n^2 - 26*n + 16)*x^3 + (3*n - 2)*x^4)*v[i - 1] - ((8*n^2 - 8*n)*x - (10*n^2 - 38*n + 16)*x^2 + (2*n^2 - 25*n + 44)*x^3 + (4*n - 18)*x^4 + 2*x^5)*deriv(v[i - 1], x))/(-n^3 + n^2 + (n^3 - 6*n^2 + 7*n)*x + (2*n^2 - 7*n)*x^2 + n*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-6fa6977267f2e73d","steps":["D+1"],"edges":["e-d0047e15fc6aaa43"],"text":"A105868 \u2192 D+1"},"headline":"A105868 \u2192 D+1","chain":null}