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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":"f42db441ee48811dda2ded5f06afe890e3076922239879ed91936e12f8812ccd","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-a8ddb226edd55d5f","role":"sequence","name":"basis_map_A013611 of Pascal-(1,2,1) array read by antidiagonals.","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:    112,     416,      352,       64;\n  4:    396,    2208,     3616,     1920,      256;\n  5:   1320,    9904,    25088,    25344,     9728,     1024;\n  6:   4264,   40288,   139520,   218624,   157184,    47104,     4096;\n  7:  13440,  153344,   677120,  1463808,  1626112,   901120,   221184,    16384;\n  8:  41616,  556800,  2995968,  8322048, 12788736, 10887168,  4890624,  1015808,   65536;\n  9: 127072, 1951296, 12398592, 42240000, 83761152, 98586624, 67633152, 25493504, 4587520, 262144;\n","recurrence_sentence":"The row polynomials satisfy (24*n^2-12*n+(48*n^3-12*n)*x+(128*n^2-160*n)*x^2+(64*n^2+16*n)*x^3+64*n*x^4)*P(n,x) = -(72*n^3-240*n^2+18*n+78+(240*n^3-864*n^2-276*n+636)*x+(160*n^2-1416*n+1304)*x^2+(320*n^2-528*n-144)*x^3+(64*n-768)*x^4-256*x^5)*P(n-1,x) + (36*n^2-9+(288*n^2+96*n-84)*x+(624*n^2+816*n-144)*x^2+(192*n^2+2048*n+656)*x^3+(1344*n+2192)*x^4+(256*n+1408)*x^5+256*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[(-(72*n^3 - 240*n^2 + 18*n + 78 + (240*n^3 - 864*n^2 - 276*n + 636)*x + (160*n^2 - 1416*n + 1304)*x^2 + (320*n^2 - 528*n - 144)*x^3 + (64*n - 768)*x^4 - 256*x^5)*P[n - 1, x] + (36*n^2 - 9 + (288*n^2 + 96*n - 84)*x + (624*n^2 + 816*n - 144)*x^2 + (192*n^2 + 2048*n + 656)*x^3 + (1344*n + 2192)*x^4 + (256*n + 1408)*x^5 + 256*x^6)*D[P[n - 1, x], x])/(24*n^2 - 12*n + (48*n^3 - 12*n)*x + (128*n^2 - 160*n)*x^2 + (64*n^2 + 16*n)*x^3 + 64*n*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] = (-(72*n^3 - 240*n^2 + 18*n + 78 + (240*n^3 - 864*n^2 - 276*n + 636)*x + (160*n^2 - 1416*n + 1304)*x^2 + (320*n^2 - 528*n - 144)*x^3 + (64*n - 768)*x^4 - 256*x^5)*v[i - 1] + (36*n^2 - 9 + (288*n^2 + 96*n - 84)*x + (624*n^2 + 816*n - 144)*x^2 + (192*n^2 + 2048*n + 656)*x^3 + (1344*n + 2192)*x^4 + (256*n + 1408)*x^5 + 256*x^6)*deriv(v[i - 1], x))/(24*n^2 - 12*n + (48*n^3 - 12*n)*x + (128*n^2 - 160*n)*x^2 + (64*n^2 + 16*n)*x^3 + 64*n*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-f25cd955a876cde3","steps":["basis A013611","D+1"],"edges":["e-84c91f2034e460dd","e-da0a786e2712b8f8"],"text":"A081577 \u2192 basis A013611 \u2192 D+1"},"headline":"A081577 \u2192 basis A013611 \u2192 D+1","chain":null}