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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":"91ce8d6754d55686a651d92134a17d5f64c8b98f5c9f610ed38c88231561ca11","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-0552ada0abe44307","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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:   19,    48,     16;\n  3:   45,   232,    288,      64;\n  4:   96,   760,   1872,    1536,     256;\n  5:  193,  2068,   7856,   12544,    7680,     1024;\n  6:  373,  5056,  26160,   64256,   75520,    36864,     4096;\n  7:  702, 11516,  75696,  253824,  456960,   423936,   172032,    16384;\n  8: 1295, 24952, 199248,  850944, 2091520,  2959360,  2265088,   786432,   65536;\n  9: 2353, 52088, 489936, 2548096, 7986432, 15412224, 17920000, 11665408, 3538944, 262144;\n","recurrence_sentence":"The row polynomials satisfy (64*n^3-176*n^2-188*n+52-(128*n^2+592*n+464)*x+(256*n^2-320*n-576)*x^2+(256*n+256)*x^3)*P(n,x) = -(128*n^3-1248*n^2+2824*n-216-(1280*n^2-4320*n-5056)*x+(512*n^2+384*n+7680)*x^2-(1536*n-1024)*x^3-2048*x^4)*P(n-1,x) + (80*n^2-300*n+25+(384*n^2-1600*n-460)*x+(256*n^2-1408*n-2784)*x^2+(1024*n-1920)*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[(-(128*n^3 - 1248*n^2 + 2824*n - 216 - (1280*n^2 - 4320*n - 5056)*x + (512*n^2 + 384*n + 7680)*x^2 - (1536*n - 1024)*x^3 - 2048*x^4)*P[n - 1, x] + (80*n^2 - 300*n + 25 + (384*n^2 - 1600*n - 460)*x + (256*n^2 - 1408*n - 2784)*x^2 + (1024*n - 1920)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*D[P[n - 1, x], x])/(64*n^3 - 176*n^2 - 188*n + 52 - (128*n^2 + 592*n + 464)*x + (256*n^2 - 320*n - 576)*x^2 + (256*n + 256)*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] = (-(128*n^3 - 1248*n^2 + 2824*n - 216 - (1280*n^2 - 4320*n - 5056)*x + (512*n^2 + 384*n + 7680)*x^2 - (1536*n - 1024)*x^3 - 2048*x^4)*v[i - 1] + (80*n^2 - 300*n + 25 + (384*n^2 - 1600*n - 460)*x + (256*n^2 - 1408*n - 2784)*x^2 + (1024*n - 1920)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*deriv(v[i - 1], x))/(64*n^3 - 176*n^2 - 188*n + 52 - (128*n^2 + 592*n + 464)*x + (256*n^2 - 320*n - 576)*x^2 + (256*n + 256)*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-21854f4257effb01","steps":["basis A013611","D+1"],"edges":["e-3723cb941a96ee35","e-c47b714461bbc6fb"],"text":"A030528 \u2192 basis A013611 \u2192 D+1"},"headline":"A030528 \u2192 basis A013611 \u2192 D+1","chain":null}