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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":"ef0632e87357cab82aa2fdef8373d4c8fe6730550c2896651738c3c09970718f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-01d7f2ed2a619101","role":"sequence","name":"basis_map_A013611 of Square the entries of Pascal's triangle.","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:      6,        4;\n  2:     30,       56,       16;\n  3:    140,      504,      384,        64;\n  4:    630,     3440,     5280,      2304,       256;\n  5:   2772,    20440,    49280,     44160,     12800,      1024;\n  6:  12012,   111888,   372960,    537600,    322560,     67584,      4096;\n  7:  51480,   580272,  2483712,   5040000,   4945920,   2150400,    344064,    16384;\n  8: 218790,  2896608, 15183168,  40094208,  56179200,  40427520,  13418496,  1703936,   65536;\n  9: 923780, 14054040, 87310080, 285212928, 525837312, 546416640, 302776320, 79626240, 8257536, 262144;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-6*n^2+2*n+(8*n^2-11*n)*x+4*n*x^2)*P(n,x) = (16*n^3-16*n^2-4*n+4+(48*n^3-40*n^2-56*n+30)*x+(96*n^2-132*n+48)*x^2+(48*n-32)*x^3)*P(n-1,x) - ((8*n^2-2)*x+(32*n^2+16*n-14)*x^2+(64*n-16)*x^3+32*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[((16*n^3 - 16*n^2 - 4*n + 4 + (48*n^3 - 40*n^2 - 56*n + 30)*x + (96*n^2 - 132*n + 48)*x^2 + (48*n - 32)*x^3)*P[n - 1, x] - ((8*n^2 - 2)*x + (32*n^2 + 16*n - 14)*x^2 + (64*n - 16)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 + 2*n + (8*n^2 - 11*n)*x + 4*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] = ((16*n^3 - 16*n^2 - 4*n + 4 + (48*n^3 - 40*n^2 - 56*n + 30)*x + (96*n^2 - 132*n + 48)*x^2 + (48*n - 32)*x^3)*v[i - 1] - ((8*n^2 - 2)*x + (32*n^2 + 16*n - 14)*x^2 + (64*n - 16)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(4*n^3 - 6*n^2 + 2*n + (8*n^2 - 11*n)*x + 4*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-b13531ad79e7c629","steps":["basis A013611","D+1"],"edges":["e-52e1bd34b019f2ca","e-e6653550c8715833"],"text":"A008459 \u2192 basis A013611 \u2192 D+1"},"headline":"A008459 \u2192 basis A013611 \u2192 D+1","chain":null}