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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":"daedc6e55e9303f8c15105c5aed64fbca12204db7d49e1ab6e3fb64161010ce3","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-efe494309672b404","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:     0;\n  2:     2;\n  3:    42,      6;\n  4:   252,    216,      12;\n  5:   924,   1980,     660,      20;\n  6:  2574,  10296,    8580,    1560,     30;\n  7:  6006,  38610,   60060,   27300,   3150,     42;\n  8: 12376, 116688,  291720,  247520,  71400,   5712,   56;\n  9: 23256, 302328, 1108536, 1511640, 813960, 162792, 9576, 72;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-9*n+10)*P(n,x) = (2*n^2+n+(6*n^2-21*n+12)*x)*P(n-1,x) + ((4*n-4)*x-(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (6*n^2 - 21*n + 12)*x)*P[n - 1, x] + ((4*n - 4)*x - (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 + n + (6*n^2 - 21*n + 12)*x)*v[i - 1] + ((4*n - 4)*x - (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10)); 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-c30cb06ad75c5e21","steps":["D","D"],"edges":["e-1287908e85a1078d","e-da2bf1668680316c"],"text":"A103327 \u2192 D \u2192 D"},"headline":"A103327 \u2192 D \u2192 D","chain":null}