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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":"55582bf92ae0d4b3413a7ea54c168d0e93b7ce439dab72cf0299bb475b0b3716","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-7196be931c2ed996","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:  3;\n  2:  4,   6;\n  3:  7,  30,   27;\n  4:  8,  44,   72,   36;\n  5: 11,  92,  270,  324,   135;\n  6: 12, 114,  408,  684,   540,   162;\n  7: 15, 186,  921, 2316,  3105,  2106,   567;\n  8: 16, 216, 1200, 3544,  6000,  5832,  3024,   648;\n  9: 19, 312, 2172, 8344, 19290, 27432, 23436, 11016, 2187;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-9*n+5+(12*n^2-33*n+21)*x+(9*n^2-27*n+18)*x^2)*P(n,x) = (12*n^2-23*n+2+(16*n^3+20*n^2-108*n+18)*x+(72*n^3-87*n^2-138*n+54)*x^2+(108*n^3-243*n^2+27*n+54)*x^3+(54*n^3-162*n^2+108*n)*x^4)*P(n-1,x) - (4*n-1+(8*n^2+26*n-10)*x+(56*n^2+46*n-36)*x^2+(138*n^2-12*n-54)*x^3+(144*n^2-90*n-27)*x^4+(54*n^2-54*n)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 23*n + 2 + (16*n^3 + 20*n^2 - 108*n + 18)*x + (72*n^3 - 87*n^2 - 138*n + 54)*x^2 + (108*n^3 - 243*n^2 + 27*n + 54)*x^3 + (54*n^3 - 162*n^2 + 108*n)*x^4)*P[n - 1, x] - (4*n - 1 + (8*n^2 + 26*n - 10)*x + (56*n^2 + 46*n - 36)*x^2 + (138*n^2 - 12*n - 54)*x^3 + (144*n^2 - 90*n - 27)*x^4 + (54*n^2 - 54*n)*x^5)*D[P[n - 1, x], x])/(4*n^2 - 9*n + 5 + (12*n^2 - 33*n + 21)*x + (9*n^2 - 27*n + 18)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^2 - 23*n + 2 + (16*n^3 + 20*n^2 - 108*n + 18)*x + (72*n^3 - 87*n^2 - 138*n + 54)*x^2 + (108*n^3 - 243*n^2 + 27*n + 54)*x^3 + (54*n^3 - 162*n^2 + 108*n)*x^4)*v[i - 1] - (4*n - 1 + (8*n^2 + 26*n - 10)*x + (56*n^2 + 46*n - 36)*x^2 + (138*n^2 - 12*n - 54)*x^3 + (144*n^2 - 90*n - 27)*x^4 + (54*n^2 - 54*n)*x^5)*deriv(v[i - 1], x))/(4*n^2 - 9*n + 5 + (12*n^2 - 33*n + 21)*x + (9*n^2 - 27*n + 18)*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-7196be931c2ed996","steps":["D"],"edges":["e-6c646dd86721ff4d"],"text":"A152842 \u2192 D"},"headline":"A152842 \u2192 D","chain":null}