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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":"ab551c036e59924b5d3b34fa3b5d932e339050290248d1ea5cf444c15e193043","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a4555bce014ca1b5","role":"sequence","name":"derivative of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    0;\n   2:    1;\n   3:    7,     2;\n   4:   28,    20,     3;\n   5:   84,   108,    39,     4;\n   6:  209,   418,   264,    64,     5;\n   7:  455,  1300,  1248,   520,    95,     6;\n   8:  896,  3456,  4650,  2900,   900,   132,    7;\n   9: 1632,  8160, 14586, 12580,  5780,  1428,  175,   8;\n  10: 2793, 17556, 40128, 45448, 28595, 10374, 2128, 224, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^4-13*n^3+47*n^2-88*n+60-(2*n^2-7*n+6)*x^2)*P(n,x) = (2*n^4-3*n^3+17*n^2-8*n+(n^4-2*n^3+5*n^2-4*n)*x-(3*n^2-2*n)*x^2-(n^2-n)*x^3)*P(n-1,x) + ((4*n^3-12*n^2+32*n-24)*x+(n^3-3*n^2+2)*x^2-(6*n-6)*x^3-(n-1)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^4 - 3*n^3 + 17*n^2 - 8*n + (n^4 - 2*n^3 + 5*n^2 - 4*n)*x - (3*n^2 - 2*n)*x^2 - (n^2 - n)*x^3)*P[n - 1, x] + ((4*n^3 - 12*n^2 + 32*n - 24)*x + (n^3 - 3*n^2 + 2)*x^2 - (6*n - 6)*x^3 - (n - 1)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 13*n^3 + 47*n^2 - 88*n + 60 - (2*n^2 - 7*n + 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n^4 - 3*n^3 + 17*n^2 - 8*n + (n^4 - 2*n^3 + 5*n^2 - 4*n)*x - (3*n^2 - 2*n)*x^2 - (n^2 - n)*x^3)*v[i - 1] + ((4*n^3 - 12*n^2 + 32*n - 24)*x + (n^3 - 3*n^2 + 2)*x^2 - (6*n - 6)*x^3 - (n - 1)*x^4)*deriv(v[i - 1], x))/(2*n^4 - 13*n^3 + 47*n^2 - 88*n + 60 - (2*n^2 - 7*n + 6)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["D","D+1"],"edges":["e-05948cabe35ede38","e-636dcd59a719bf9c"],"text":"A111125 \u2192 D \u2192 D+1"},"headline":"A111125 \u2192 D \u2192 D+1","chain":null}