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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":"e091cc53e8f2d1794adb551c97b921312911b0bc1e566b4cb32341925b9a8b16","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-373413a99f5cc9e6","role":"sequence","name":"veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","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:      0;\n  1:      1;\n  2:      8;\n  3:     49,      1;\n  4:    272,     16;\n  5:   1440,    162,     1;\n  6:   7424,   1328,    24;\n  7:  37632,   9632,   339,   1;\n  8: 188416,  64512,  3680,  32;\n  9: 933888, 408576, 33984, 580, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-24*n^2+428*n-408+(16*n^2-328*n+312)*x+(16*n-16)*x^2)*P(n,x) = (16*n^3-48*n^2+1568*n+(n^3+57*n^2-1202*n-72)*x+(4*n^2-18*n+148)*x^2+(4*n-8)*x^3)*P(n-1,x) + ((32*n^2-96*n+1088)*x-(2*n^2-134*n+2308)*x^2-(8*n-268)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 48*n^2 + 1568*n + (n^3 + 57*n^2 - 1202*n - 72)*x + (4*n^2 - 18*n + 148)*x^2 + (4*n - 8)*x^3)*P[n - 1, x] + ((32*n^2 - 96*n + 1088)*x - (2*n^2 - 134*n + 2308)*x^2 - (8*n - 268)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 24*n^2 + 428*n - 408 + (16*n^2 - 328*n + 312)*x + (16*n - 16)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^3 - 48*n^2 + 1568*n + (n^3 + 57*n^2 - 1202*n - 72)*x + (4*n^2 - 18*n + 148)*x^2 + (4*n - 8)*x^3)*v[i - 1] + ((32*n^2 - 96*n + 1088)*x - (2*n^2 - 134*n + 2308)*x^2 - (8*n - 268)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 24*n^2 + 428*n - 408 + (16*n^2 - 328*n + 312)*x + (16*n - 16)*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-3dcaebc50e901d48","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-4a3fbc77a3e37655","e-6714daf00f32273e"],"text":"A038231 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A038231 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}