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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":"093bda2852dcd99985876bfc80cda656e7ed914c126604210782549102c57c1e","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-0c97d42b0cd05e48","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.","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:      2,        1;\n  2:      7,        8,         1;\n  3:     28,       69,        24,          1;\n  4:    113,      652,       444,         62,          1;\n  5:    454,     6209,      8092,       2302,        146,         1;\n  6:   1819,    58152,    146127,      74276,      10329,       324,        1;\n  7:   7280,   536101,   2574276,    2272975,     564080,     41775,      692,      1;\n  8:  29125,  4890428,  44147890,   66880346,   27538450,   3738320,   156598,   1442,    1;\n  9: 116506, 44334225, 740628360, 1897609404, 1260229776, 280844490, 22337472, 555048, 2958, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-4+4*x)*P(n,x) = -(4*n-12+(2*n^2-15*n+43)*x+(n^3-10*n^2+37*n-60)*x^2+(6*n^2-36*n+38)*x^3+(8*n-8)*x^4)*P(n-2,x) - ((5*n-15)*x+(4*n^2-29*n+71)*x^2-(n^2-29*n+98)*x^3-(6*n-46)*x^4-8*x^5)*d/dx P(n-2,x) - ((n-3)*x^2+(n^2-7*n+16)*x^3+(6*n-22)*x^4+8*x^5)*d^2/dx^2 P(n-2,x) + (5*n-19+(2*n^2-15*n+48)*x+(10*n-38)*x^2+8*x^3)*P(n-1,x) + ((5*n-20)*x-(n-24)*x^2-4*x^3)*d/dx P(n-1,x) + ((n-4)*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 12 + (2*n^2 - 15*n + 43)*x + (n^3 - 10*n^2 + 37*n - 60)*x^2 + (6*n^2 - 36*n + 38)*x^3 + (8*n - 8)*x^4)*P[n - 2, x] - ((5*n - 15)*x + (4*n^2 - 29*n + 71)*x^2 - (n^2 - 29*n + 98)*x^3 - (6*n - 46)*x^4 - 8*x^5)*D[P[n - 2, x], x] - ((n - 3)*x^2 + (n^2 - 7*n + 16)*x^3 + (6*n - 22)*x^4 + 8*x^5)*D[P[n - 2, x], {x, 2}] + (5*n - 19 + (2*n^2 - 15*n + 48)*x + (10*n - 38)*x^2 + 8*x^3)*P[n - 1, x] + ((5*n - 20)*x - (n - 24)*x^2 - 4*x^3)*D[P[n - 1, x], x] + ((n - 4)*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}])/(n - 4 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 12 + (2*n^2 - 15*n + 43)*x + (n^3 - 10*n^2 + 37*n - 60)*x^2 + (6*n^2 - 36*n + 38)*x^3 + (8*n - 8)*x^4)*v[i - 2] - ((5*n - 15)*x + (4*n^2 - 29*n + 71)*x^2 - (n^2 - 29*n + 98)*x^3 - (6*n - 46)*x^4 - 8*x^5)*deriv(v[i - 2], x) - ((n - 3)*x^2 + (n^2 - 7*n + 16)*x^3 + (6*n - 22)*x^4 + 8*x^5)*deriv(deriv(v[i - 2], x), x) + (5*n - 19 + (2*n^2 - 15*n + 48)*x + (10*n - 38)*x^2 + 8*x^3)*v[i - 1] + ((5*n - 20)*x - (n - 24)*x^2 - 4*x^3)*deriv(v[i - 1], x) + ((n - 4)*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 4 + 4*x)); 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-0c97d42b0cd05e48","steps":["D+1"],"edges":["e-0dbdea355067bcdb"],"text":"A166960 \u2192 D+1"},"headline":"A166960 \u2192 D+1","chain":null}