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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":"2c3c5c33a517eafd42440b554f85c630a1b28244a1238d17738d3f35dc638519","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-f8c44f01f6cd4a34","role":"sequence","name":"basis_map_A013609 of Riordan array (1, 3+3x).","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,        6;\n  2:      54,      114,        36;\n  3:     279,     1026,      1044,       216;\n  4:    1341,     7074,     12672,      8424,       1296;\n  5:    6156,    42444,    109836,    128952,      63504,       7776;\n  6:   27405,   233604,    793152,   1349352,    1174176,     458784,      46656;\n  7:  119313,  1212246,   5102028,  11413224,   14410224,    9945504,    3219264,    279936;\n  8:  510786,  6027534,  30293352,  84138264,  139852656,  140115744,   80061696,  22114944,   1679616;\n  9: 2158083, 29012742, 169587756, 563163192, 1163390688, 1535417856, 1273662144, 620618112, 149485824, 10077696;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-4*n^2-22*n+(32*n^2-76*n)*x+(48*n^2-40*n)*x^2+48*n*x^3)*P(n,x) = -(16*n^3-128*n^2+128*n+146-(80*n^2-96*n-488)*x+(96*n^2-224*n+320)*x^2-(192*n+288)*x^3-288*x^4)*P(n-1,x) + (28*n^2-14*n-21+(80*n^2+72*n-46)*x+(48*n^2+464*n+200)*x^2+(672*n+752)*x^3+(288*n+816)*x^4+288*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[(-(16*n^3 - 128*n^2 + 128*n + 146 - (80*n^2 - 96*n - 488)*x + (96*n^2 - 224*n + 320)*x^2 - (192*n + 288)*x^3 - 288*x^4)*P[n - 1, x] + (28*n^2 - 14*n - 21 + (80*n^2 + 72*n - 46)*x + (48*n^2 + 464*n + 200)*x^2 + (672*n + 752)*x^3 + (288*n + 816)*x^4 + 288*x^5)*D[P[n - 1, x], x])/(8*n^3 - 4*n^2 - 22*n + (32*n^2 - 76*n)*x + (48*n^2 - 40*n)*x^2 + 48*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] = (-(16*n^3 - 128*n^2 + 128*n + 146 - (80*n^2 - 96*n - 488)*x + (96*n^2 - 224*n + 320)*x^2 - (192*n + 288)*x^3 - 288*x^4)*v[i - 1] + (28*n^2 - 14*n - 21 + (80*n^2 + 72*n - 46)*x + (48*n^2 + 464*n + 200)*x^2 + (672*n + 752)*x^3 + (288*n + 816)*x^4 + 288*x^5)*deriv(v[i - 1], x))/(8*n^3 - 4*n^2 - 22*n + (32*n^2 - 76*n)*x + (48*n^2 - 40*n)*x^2 + 48*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-016d7724d51f6416","steps":["basis A013609","D+1"],"edges":["e-2339e629d7f9e0b9","e-bcfc17226a30a42f"],"text":"A099093 \u2192 basis A013609 \u2192 D+1"},"headline":"A099093 \u2192 basis A013609 \u2192 D+1","chain":null}