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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":"6efc07699b158e7b0ac2fd3c3e00729a645b17e1a041084772eac72faa0c4b77","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-f074ae9ffad82cf4","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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:    1;\n  2:   10,     6;\n  3:   19,    12;\n  4:   73,   126,    36;\n  5:  172,   348,   108;\n  6:  505,  1488,  1188,   216;\n  7: 1261,  4356,  4248,   864;\n  8: 3358, 14298, 19980,  9936, 1296;\n  9: 8443, 41196, 68796, 43200, 6480;\n","recurrence_sentence":"The row polynomials satisfy (24*n^3-20*n^2-50*n+(96*n^2-140*n)*x+96*n*x^2)*P(n,x) = (168*n^3-260*n^2-230*n+276+(288*n^3+144*n^2-1292*n+816)*x+(1152*n^2-1008*n-48)*x^2+(1152*n-1152)*x^3)*P(n-1,x) - (156*n^2+182*n+13+(600*n^2+1324*n+388)*x+(576*n^2+3072*n+1972)*x^2+(2304*n+3648)*x^3+2304*x^4)*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[((168*n^3 - 260*n^2 - 230*n + 276 + (288*n^3 + 144*n^2 - 1292*n + 816)*x + (1152*n^2 - 1008*n - 48)*x^2 + (1152*n - 1152)*x^3)*P[n - 1, x] - (156*n^2 + 182*n + 13 + (600*n^2 + 1324*n + 388)*x + (576*n^2 + 3072*n + 1972)*x^2 + (2304*n + 3648)*x^3 + 2304*x^4)*D[P[n - 1, x], x])/(24*n^3 - 20*n^2 - 50*n + (96*n^2 - 140*n)*x + 96*n*x^2)]];\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] = ((168*n^3 - 260*n^2 - 230*n + 276 + (288*n^3 + 144*n^2 - 1292*n + 816)*x + (1152*n^2 - 1008*n - 48)*x^2 + (1152*n - 1152)*x^3)*v[i - 1] - (156*n^2 + 182*n + 13 + (600*n^2 + 1324*n + 388)*x + (576*n^2 + 3072*n + 1972)*x^2 + (2304*n + 3648)*x^3 + 2304*x^4)*deriv(v[i - 1], x))/(24*n^3 - 20*n^2 - 50*n + (96*n^2 - 140*n)*x + 96*n*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-e189038e95d51606","steps":["basis A013609","D+1"],"edges":["e-79f6722e524950c5","e-1bbf3d440b74ff15"],"text":"A304236 \u2192 basis A013609 \u2192 D+1"},"headline":"A304236 \u2192 basis A013609 \u2192 D+1","chain":null}