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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":"43b29c1f2d3b459b29a68fa69bfe12e5ec75105d5061ee302dc1034a0a9ecac7","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-1dda68ade0f78df3","role":"sequence","name":"Triangle read by rows of Legendre-Stirling numbers of the second kind.","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:   8,     8;\n  3:  20,   104,      24;\n  4:  40,   584,     960,       64;\n  5:  70,  2184,   11472,     7744,       160;\n  6: 112,  6384,   76992,   191296,     58240,        384;\n  7: 168, 15792,  364272,  2244416,   2927680,     419712,       896;\n  8: 240, 34608, 1359168, 16815296,  59038080,   42578304,   2938880,     2048;\n  9: 330, 69168, 4266240, 92928704, 689611680, 1459492224, 599035136, 20154368, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n+1+(2*n^3-4*n^2-2*n+4)*x+(n^5-3*n^4+n^3+3*n^2-2*n)*x^2)*P(n-2,x) + ((2*n^2-3*n-5)*x^2+(2*n^4-4*n^3-2*n^2+4*n)*x^3)*d/dx P(n-2,x) - ((n+1)*x^3+(n^3-n)*x^4)*d^2/dx^2 P(n-2,x) + (2*n+(2*n^3-4*n^2+2*n)*x)*P(n-1,x) - (2*n^2-5*n+3)*x^2*d/dx P(n-1,x) + (n-1)*x^3*d^2/dx^2 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[(-(n + 1 + (2*n^3 - 4*n^2 - 2*n + 4)*x + (n^5 - 3*n^4 + n^3 + 3*n^2 - 2*n)*x^2)*P[n - 2, x] + ((2*n^2 - 3*n - 5)*x^2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] - ((n + 1)*x^3 + (n^3 - n)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n + (2*n^3 - 4*n^2 + 2*n)*x)*P[n - 1, x] - (2*n^2 - 5*n + 3)*x^2*D[P[n - 1, x], x] + (n - 1)*x^3*D[P[n - 1, x], {x, 2}])/(n - 1)]];\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] = (-(n + 1 + (2*n^3 - 4*n^2 - 2*n + 4)*x + (n^5 - 3*n^4 + n^3 + 3*n^2 - 2*n)*x^2)*v[i - 2] + ((2*n^2 - 3*n - 5)*x^2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) - ((n + 1)*x^3 + (n^3 - n)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n + (2*n^3 - 4*n^2 + 2*n)*x)*v[i - 1] - (2*n^2 - 5*n + 3)*x^2*deriv(v[i - 1], x) + (n - 1)*x^3*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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-1dda68ade0f78df3","steps":["D"],"edges":["e-587213a35d2f6866"],"text":"A191935 \u2192 D"},"headline":"A191935 \u2192 D","chain":null}