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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":"c5e120ea44fedb8d92631d0715ce25925d2b446e1e1c6f9d96c06fbd9e1eb33d","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-20086f90792ae22f","role":"sequence","name":"basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:      9,       4;\n  2:     41,      40,       8;\n  3:    185,     284,     128,       16;\n  4:    793,    1680,    1224,      352,      32;\n  5:   3305,    8964,    9120,     4272,     896,      64;\n  6:  13481,   44664,   58632,    38592,   13216,    2176,     128;\n  7:  54105,  212012,  341760,   291536,  140672,   37824,    5120,    256;\n  8: 214393,  970592, 1857544,  1952288, 1223840,  464128,  102528,  11776,   512;\n  9: 840841, 4320340, 9580192, 11985136, 9275776, 4575808, 1425920, 267008, 26624, 1024;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+17*n^2-34*n+5+(8*n^3+54*n^2-59*n-33)*x+(44*n^2-2*n-74)*x^2+(8*n^2+28*n-44)*x^3+(8*n-8)*x^4)*P(n,x) = -(28*n^3+67*n^2-18*n-5+(24*n^3+154*n^2+91*n+33)*x+(116*n^2+218*n+74)*x^2+(24*n^2+132*n+44)*x^3+(24*n+8)*x^4)*P(n-1,x) + (68*n^2+357*n+408+(148*n^2+1117*n+1619)*x+(96*n^2+1312*n+2575)*x^2+(16*n^2+712*n+2084)*x^3+(176*n+896)*x^4+(16*n+192)*x^5+16*x^6)*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[(-(28*n^3 + 67*n^2 - 18*n - 5 + (24*n^3 + 154*n^2 + 91*n + 33)*x + (116*n^2 + 218*n + 74)*x^2 + (24*n^2 + 132*n + 44)*x^3 + (24*n + 8)*x^4)*P[n - 1, x] + (68*n^2 + 357*n + 408 + (148*n^2 + 1117*n + 1619)*x + (96*n^2 + 1312*n + 2575)*x^2 + (16*n^2 + 712*n + 2084)*x^3 + (176*n + 896)*x^4 + (16*n + 192)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(4*n^3 + 17*n^2 - 34*n + 5 + (8*n^3 + 54*n^2 - 59*n - 33)*x + (44*n^2 - 2*n - 74)*x^2 + (8*n^2 + 28*n - 44)*x^3 + (8*n - 8)*x^4)]];\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] = (-(28*n^3 + 67*n^2 - 18*n - 5 + (24*n^3 + 154*n^2 + 91*n + 33)*x + (116*n^2 + 218*n + 74)*x^2 + (24*n^2 + 132*n + 44)*x^3 + (24*n + 8)*x^4)*v[i - 1] + (68*n^2 + 357*n + 408 + (148*n^2 + 1117*n + 1619)*x + (96*n^2 + 1312*n + 2575)*x^2 + (16*n^2 + 712*n + 2084)*x^3 + (176*n + 896)*x^4 + (16*n + 192)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(4*n^3 + 17*n^2 - 34*n + 5 + (8*n^3 + 54*n^2 - 59*n - 33)*x + (44*n^2 - 2*n - 74)*x^2 + (8*n^2 + 28*n - 44)*x^3 + (8*n - 8)*x^4)); 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-e646c30d067149e7","steps":["basis A007318","D+1"],"edges":["e-4395b7c136181822","e-8f285cbc321bca86"],"text":"A249252 \u2192 basis A007318 \u2192 D+1"},"headline":"A249252 \u2192 basis A007318 \u2192 D+1","chain":null}