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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":"f19ab4ee030141403e916a9f8b144bbe0add6d9af709e005e672dcecff0706bb","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-3a1bf08f9f4a4163","role":"sequence","name":"basis_map_A013610 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:     5,       3;\n  2:    24,      36,       9;\n  3:    88,     252,     171,       27;\n  4:   308,    1308,    1656,      702,       81;\n  5:  1020,    5796,   11196,     8748,     2673,      243;\n  6:  3280,   23400,   61416,    73440,    40824,     9720,     729;\n  7: 10304,   88608,  295560,   484488,   410832,   175932,   34263,    2187;\n  8: 31824,  320544, 1300320,  2729376,  3181356,  2067444,  717336,  118098,   6561;\n  9: 96976, 1120176, 5359392, 13768704, 20636532, 18426204, 9649044, 2808108, 400221, 19683;\n","recurrence_sentence":"The row polynomials satisfy (18*n^2-6*n+(27*n^3+9*n^2-6*n)*x+(81*n^2-81*n)*x^2+(27*n^2+36*n)*x^3+27*n*x^4)*P(n,x) = -(54*n^3-162*n^2-24*n+72+(135*n^3-387*n^2-354*n+420)*x+(216*n^2-846*n+540)*x^2+(135*n^2-117*n-333)*x^3+(54*n-378)*x^4-81*x^5)*P(n-1,x) + (36*n^2+12*n-8+(216*n^2+180*n-36)*x+(351*n^2+801*n+114)*x^2+(81*n^2+1296*n+855)*x^3+(594*n+1413)*x^4+(81*n+621)*x^5+81*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[(-(54*n^3 - 162*n^2 - 24*n + 72 + (135*n^3 - 387*n^2 - 354*n + 420)*x + (216*n^2 - 846*n + 540)*x^2 + (135*n^2 - 117*n - 333)*x^3 + (54*n - 378)*x^4 - 81*x^5)*P[n - 1, x] + (36*n^2 + 12*n - 8 + (216*n^2 + 180*n - 36)*x + (351*n^2 + 801*n + 114)*x^2 + (81*n^2 + 1296*n + 855)*x^3 + (594*n + 1413)*x^4 + (81*n + 621)*x^5 + 81*x^6)*D[P[n - 1, x], x])/(18*n^2 - 6*n + (27*n^3 + 9*n^2 - 6*n)*x + (81*n^2 - 81*n)*x^2 + (27*n^2 + 36*n)*x^3 + 27*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] = (-(54*n^3 - 162*n^2 - 24*n + 72 + (135*n^3 - 387*n^2 - 354*n + 420)*x + (216*n^2 - 846*n + 540)*x^2 + (135*n^2 - 117*n - 333)*x^3 + (54*n - 378)*x^4 - 81*x^5)*v[i - 1] + (36*n^2 + 12*n - 8 + (216*n^2 + 180*n - 36)*x + (351*n^2 + 801*n + 114)*x^2 + (81*n^2 + 1296*n + 855)*x^3 + (594*n + 1413)*x^4 + (81*n + 621)*x^5 + 81*x^6)*deriv(v[i - 1], x))/(18*n^2 - 6*n + (27*n^3 + 9*n^2 - 6*n)*x + (81*n^2 - 81*n)*x^2 + (27*n^2 + 36*n)*x^3 + 27*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 A013610","D+1"],"edges":["e-ce7b55dc0173f12e","e-e7cf9b252993f380"],"text":"A081577 \u2192 basis A013610 \u2192 D+1"},"headline":"A081577 \u2192 basis A013610 \u2192 D+1","chain":null}