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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":"abfad559872bd94c0dfc844d87abba5796aad16dc6138a10bc5a26de13611b29","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-b2e2c5159c22bcc9","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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:     11,       8;\n  2:     44,     100,       32;\n  3:    153,     612,      624,      128;\n  4:    499,    2876,     5248,     3392,      512;\n  5:   1564,   11788,    31488,    36416,    17152,     2048;\n  6:   4769,   44424,   157104,   266880,   224256,    82944,     8192;\n  7:  14251,  158128,   697584,  1575424,  1944320,  1278976,   389120,    32768;\n  8:  41932,  540020,  2857888,  8083264, 13277440, 12813312,  6914048,  1785856,  131072;\n  9: 121881, 1787052, 11037504, 37582656, 77435136, 99449856, 78618624, 35930112, 8060928, 524288;\n","recurrence_sentence":"The row polynomials satisfy (128*n^3-960*n^2+728*n-28+(1024*n^3-3072*n^2+2304*n-352)*x-(640*n-832)*x^2-(1024*n-1024)*x^3)*P(n,x) = (192*n^3-1344*n^2-812*n+512+(1792*n^3-4096*n^2-3408*n+3600)*x+(2048*n^3-2048*n^2-5824*n+7168)*x^2-(4864*n-6400)*x^3-(2048*n-2048)*x^4)*P(n-1,x) + (80*n^2-320*n-505+(1024*n^2-2176*n-4024)*x+(3328*n^2-4096*n-10976)*x^2+(2048*n^2-2048*n-13824)*x^3-8448*x^4-2048*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[((192*n^3 - 1344*n^2 - 812*n + 512 + (1792*n^3 - 4096*n^2 - 3408*n + 3600)*x + (2048*n^3 - 2048*n^2 - 5824*n + 7168)*x^2 - (4864*n - 6400)*x^3 - (2048*n - 2048)*x^4)*P[n - 1, x] + (80*n^2 - 320*n - 505 + (1024*n^2 - 2176*n - 4024)*x + (3328*n^2 - 4096*n - 10976)*x^2 + (2048*n^2 - 2048*n - 13824)*x^3 - 8448*x^4 - 2048*x^5)*D[P[n - 1, x], x])/(128*n^3 - 960*n^2 + 728*n - 28 + (1024*n^3 - 3072*n^2 + 2304*n - 352)*x - (640*n - 832)*x^2 - (1024*n - 1024)*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] = ((192*n^3 - 1344*n^2 - 812*n + 512 + (1792*n^3 - 4096*n^2 - 3408*n + 3600)*x + (2048*n^3 - 2048*n^2 - 5824*n + 7168)*x^2 - (4864*n - 6400)*x^3 - (2048*n - 2048)*x^4)*v[i - 1] + (80*n^2 - 320*n - 505 + (1024*n^2 - 2176*n - 4024)*x + (3328*n^2 - 4096*n - 10976)*x^2 + (2048*n^2 - 2048*n - 13824)*x^3 - 8448*x^4 - 2048*x^5)*deriv(v[i - 1], x))/(128*n^3 - 960*n^2 + 728*n - 28 + (1024*n^3 - 3072*n^2 + 2304*n - 352)*x - (640*n - 832)*x^2 - (1024*n - 1024)*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-353fb5a029e4e669","steps":["basis A013611","D+1"],"edges":["e-a3f64a2b3fc7c9ce","e-2f95abe73862e45d"],"text":"A207607 \u2192 basis A013611 \u2192 D+1"},"headline":"A207607 \u2192 basis A013611 \u2192 D+1","chain":null}