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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":"12a96dba3c8dfed8d3280b1df8bba36824cb8a877a17eeb371792286b3746daa","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-ee693a689121a12c","role":"sequence","name":"Triangle of numbers used for basis change between certain falling factorials.","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:   5,     4;\n   3:  13,    72,      30;\n   4:  25,   360,    1176,      336;\n   5:  41,  1120,    9180,    23040,       5040;\n   6:  61,  2700,   40920,   250800,     534600,       95040;\n   7:  85,  5544,  133770,  1485120,    7567560,    14414400,      2162160;\n   8: 113, 10192,  357840,  6232800,   56565600,   253169280,    443963520,     57657600;\n   9: 145, 17280,  831096, 20837376,  293025600,  2303078400,   9356256000,  15397724160,   1764322560;\n  10: 181, 27540, 1737360, 59179680, 1186753680, 14299649280, 100800806400, 379761177600, 594255916800, 60949324800;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-6*n+5+(3*n-5)*x+x^2)*P(n,x) = (2*n^2-2*n+1+(8*n^3-22*n^2+21*n-9)*x+(2*n^4-12*n^3+35*n^2-43*n+19)*x^2+(3*n^3-12*n^2+19*n-10)*x^3+(n^2-3*n+2)*x^4)*P(n-1,x) - ((2*n^2-2*n+1)*x^2+(2*n^3-6*n^2+8*n-4)*x^3+(3*n^2-5*n+3)*x^4+(n-1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 2*n + 1 + (8*n^3 - 22*n^2 + 21*n - 9)*x + (2*n^4 - 12*n^3 + 35*n^2 - 43*n + 19)*x^2 + (3*n^3 - 12*n^2 + 19*n - 10)*x^3 + (n^2 - 3*n + 2)*x^4)*P[n - 1, x] - ((2*n^2 - 2*n + 1)*x^2 + (2*n^3 - 6*n^2 + 8*n - 4)*x^3 + (3*n^2 - 5*n + 3)*x^4 + (n - 1)*x^5)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 5 + (3*n - 5)*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 2*n + 1 + (8*n^3 - 22*n^2 + 21*n - 9)*x + (2*n^4 - 12*n^3 + 35*n^2 - 43*n + 19)*x^2 + (3*n^3 - 12*n^2 + 19*n - 10)*x^3 + (n^2 - 3*n + 2)*x^4)*v[i - 1] - ((2*n^2 - 2*n + 1)*x^2 + (2*n^3 - 6*n^2 + 8*n - 4)*x^3 + (3*n^2 - 5*n + 3)*x^4 + (n - 1)*x^5)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 5 + (3*n - 5)*x + x^2)); 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-ee693a689121a12c","steps":["D+1"],"edges":["e-771c5351a2234324"],"text":"A089503 \u2192 D+1"},"headline":"A089503 \u2192 D+1","chain":null}