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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":"f49b98ad89ad447e3fa595b4625bb5387c5fe49d574e19dc89b8fef39aa325bb","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-e4013d25fc6715f3","role":"sequence","name":"gamma_polynomial of Pascal-(1,8,1) array.","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:  1;\n  2:  9,    8;\n  3: 17,   16;\n  4: 25,  152,    64;\n  5: 33,  416,   192;\n  6: 41,  808,  1920,   512;\n  7: 49, 1328,  6784,  2048;\n  8: 57, 1976, 16320, 21504,  4096;\n  9: 65, 2752, 32064, 92160, 20480;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-23*n^2+15*n+(32*n^2-79*n)*x+32*n*x^2)*P(n,x) = (8*n^3-15*n^2+7*n+(128*n^3-464*n^2+289*n)*x+(512*n^2-1232*n+752)*x^2+(512*n-512)*x^3)*P(n-1,x) - ((8*n^2-7*n)*x+(256*n^2-192*n-15)*x^2+(1024*n-448)*x^3+1024*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[((8*n^3 - 15*n^2 + 7*n + (128*n^3 - 464*n^2 + 289*n)*x + (512*n^2 - 1232*n + 752)*x^2 + (512*n - 512)*x^3)*P[n - 1, x] - ((8*n^2 - 7*n)*x + (256*n^2 - 192*n - 15)*x^2 + (1024*n - 448)*x^3 + 1024*x^4)*D[P[n - 1, x], x])/(8*n^3 - 23*n^2 + 15*n + (32*n^2 - 79*n)*x + 32*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] = ((8*n^3 - 15*n^2 + 7*n + (128*n^3 - 464*n^2 + 289*n)*x + (512*n^2 - 1232*n + 752)*x^2 + (512*n - 512)*x^3)*v[i - 1] - ((8*n^2 - 7*n)*x + (256*n^2 - 192*n - 15)*x^2 + (1024*n - 448)*x^3 + 1024*x^4)*deriv(v[i - 1], x))/(8*n^3 - 23*n^2 + 15*n + (32*n^2 - 79*n)*x + 32*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-1c472d1762258215","steps":["\u03b3","D+1"],"edges":["e-023010f9acf2a31f","e-958cb68a8d390364"],"text":"A143683 \u2192 \u03b3 \u2192 D+1"},"headline":"A143683 \u2192 \u03b3 \u2192 D+1","chain":null}