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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":"c9bdf740b4826a5f5dc5f9b186bccd2066f4f0e12eee3610c8643e62969e33ee","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-096818d27dad5765","role":"sequence","name":"basis_map_A013609 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:     10,      6;\n  2:     34,     48,      12;\n  3:    124,    288,     168,      24;\n  4:    412,   1344,    1368,     480,      48;\n  5:   1336,   5640,    8304,    5088,    1248,      96;\n  6:   4216,  21888,   42528,   38400,   16416,    3072,     192;\n  7:  13072,  80640,  195264,  235584,  149184,   48384,    7296,    384;\n  8:  39952, 285696,  830016, 1260288, 1074240,  516096,  134016,  16896,   768;\n  9: 120736, 982368, 3332736, 6117120, 6604416, 4284288, 1644288, 354816, 38400, 1536;\n","recurrence_sentence":"The row polynomials satisfy -(3*n-3-(3*n^3+6*n^2-15*n+3)*x-(12*n^2-10*n-11)*x^2-(2*n^2+7*n-11)*x^3-(2*n-2)*x^4)*P(n,x) = -(9*n^3+9*n^2-15*n-3+(15*n^3+39*n^2-21*n-3)*x+(54*n^2+43*n+11)*x^2+(10*n^2+53*n+11)*x^3+(10*n+2)*x^4)*P(n-1,x) + (9*n^2+27*n+18+(36*n^2+144*n+123)*x+(39*n^2+267*n+321)*x^2+(6*n^2+198*n+395)*x^3+(50*n+227)*x^4+(4*n+52)*x^5+4*x^6)*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[(-(9*n^3 + 9*n^2 - 15*n - 3 + (15*n^3 + 39*n^2 - 21*n - 3)*x + (54*n^2 + 43*n + 11)*x^2 + (10*n^2 + 53*n + 11)*x^3 + (10*n + 2)*x^4)*P[n - 1, x] + (9*n^2 + 27*n + 18 + (36*n^2 + 144*n + 123)*x + (39*n^2 + 267*n + 321)*x^2 + (6*n^2 + 198*n + 395)*x^3 + (50*n + 227)*x^4 + (4*n + 52)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(-3*n + 3 + (3*n^3 + 6*n^2 - 15*n + 3)*x + (12*n^2 - 10*n - 11)*x^2 + (2*n^2 + 7*n - 11)*x^3 + (2*n - 2)*x^4)]];\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] = (-(9*n^3 + 9*n^2 - 15*n - 3 + (15*n^3 + 39*n^2 - 21*n - 3)*x + (54*n^2 + 43*n + 11)*x^2 + (10*n^2 + 53*n + 11)*x^3 + (10*n + 2)*x^4)*v[i - 1] + (9*n^2 + 27*n + 18 + (36*n^2 + 144*n + 123)*x + (39*n^2 + 267*n + 321)*x^2 + (6*n^2 + 198*n + 395)*x^3 + (50*n + 227)*x^4 + (4*n + 52)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(-3*n + 3 + (3*n^3 + 6*n^2 - 15*n + 3)*x + (12*n^2 - 10*n - 11)*x^2 + (2*n^2 + 7*n - 11)*x^3 + (2*n - 2)*x^4)); 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-eac80c2bf8e566f5","steps":["basis A013609","D+1"],"edges":["e-509e22279eb55844","e-4a82635fc1473374"],"text":"A249250 \u2192 basis A013609 \u2192 D+1"},"headline":"A249250 \u2192 basis A013609 \u2192 D+1","chain":null}