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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":"099914f5d6845e31ce0a81159588b643bd60798aac1c3fa9bc60ba77777ff473","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-8588fa6ee4014167","role":"sequence","name":"basis_map_A013611 of Riordan array (1, 2+2x).","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:     10,       8;\n  2:     46,     168,       64;\n  3:    176,    1184,     2048,       512;\n  4:    620,    6080,    19648,     22016,       4096;\n  5:   2072,   26784,   127616,    267776,     221184,      32768;\n  6:   6696,  107488,   678784,   2109952,    3260416,    2129920,     262144;\n  7:  21120,  405248,  3190784,  13189120,   30244864,   36896768,   19922944,    2097152;\n  8:  65424, 1460736, 13785600,  71225344,  217583616,  394067968,  396623872,  182452224,   16777216;\n  9: 199840, 5089920, 56011776, 347533312, 1329078272, 3209461760, 4794613760, 4104126464, 1644167168, 134217728;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-24*n^2-10*n-104*n*x+(128*n^2-224*n)*x^2+128*n*x^3)*P(n,x) = -(32*n^3-240*n^2+268*n+120-(384*n^2-752*n-816)*x+(256*n^2+64*n+1472)*x^2-(768*n-256)*x^3-1024*x^4)*P(n-1,x) + (24*n^2-36*n-15+(128*n^2-192*n-140)*x+(128*n^2-352)*x^2+(1024*n+128)*x^3+(1024*n+1280)*x^4+1024*x^5)*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[(-(32*n^3 - 240*n^2 + 268*n + 120 - (384*n^2 - 752*n - 816)*x + (256*n^2 + 64*n + 1472)*x^2 - (768*n - 256)*x^3 - 1024*x^4)*P[n - 1, x] + (24*n^2 - 36*n - 15 + (128*n^2 - 192*n - 140)*x + (128*n^2 - 352)*x^2 + (1024*n + 128)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*D[P[n - 1, x], x])/(16*n^3 - 24*n^2 - 10*n - 104*n*x + (128*n^2 - 224*n)*x^2 + 128*n*x^3)]];\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] = (-(32*n^3 - 240*n^2 + 268*n + 120 - (384*n^2 - 752*n - 816)*x + (256*n^2 + 64*n + 1472)*x^2 - (768*n - 256)*x^3 - 1024*x^4)*v[i - 1] + (24*n^2 - 36*n - 15 + (128*n^2 - 192*n - 140)*x + (128*n^2 - 352)*x^2 + (1024*n + 128)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*deriv(v[i - 1], x))/(16*n^3 - 24*n^2 - 10*n - 104*n*x + (128*n^2 - 224*n)*x^2 + 128*n*x^3)); 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-f5a001953ee28e8f","steps":["basis A013611","D+1"],"edges":["e-5d40644b540aa393","e-1c44dfb2daff0896"],"text":"A099040 \u2192 basis A013611 \u2192 D+1"},"headline":"A099040 \u2192 basis A013611 \u2192 D+1","chain":null}