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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":"697fefd649d837a4a5a3d079481d030eefaa2ede0e8c408cb0e294912d7d4c0c","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-a0e88202514b92bb","role":"sequence","name":"reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:  0;\n  1:  2;\n  2:  5,   8;\n  3:  8,  32,   24;\n  4: 11,  74,  132,    64;\n  5: 14, 134,  402,   448,   160;\n  6: 17, 212,  915,  1696,  1360,   384;\n  7: 20, 308, 1752,  4672,  6160,  3840,   896;\n  8: 23, 422, 2994, 10564, 19960, 20256, 10304,  2048;\n  9: 26, 554, 4722, 20884, 52210, 75552, 62048, 26624, 4608;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^2-7*n+4-(6*n^2+2*n-8)*x+(8*n-8)*x^2)*P(n,x) = -(3*n^2-4*n+1+(12*n^2-50*n+10)*x-(52*n-44)*x^2+32*x^3)*P(n-1,x) + ((3*n-1)*x+(24*n-12)*x^2+(12*n-36)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n^2 - 4*n + 1 + (12*n^2 - 50*n + 10)*x - (52*n - 44)*x^2 + 32*x^3)*P[n - 1, x] + ((3*n - 1)*x + (24*n - 12)*x^2 + (12*n - 36)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(-3*n^2 + 7*n - 4 + (6*n^2 + 2*n - 8)*x - (8*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n^2 - 4*n + 1 + (12*n^2 - 50*n + 10)*x - (52*n - 44)*x^2 + 32*x^3)*v[i - 1] + ((3*n - 1)*x + (24*n - 12)*x^2 + (12*n - 36)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(-3*n^2 + 7*n - 4 + (6*n^2 + 2*n - 8)*x - (8*n - 8)*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-d5c14b9fec875ecc","steps":["reverse","D"],"edges":["e-635d5bcbc56b7dbf","e-532da91c70c88463"],"text":"A121574 \u2192 reverse \u2192 D"},"headline":"A121574 \u2192 reverse \u2192 D","chain":null}