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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":"0e85b0a25dda7ce6684c370e830a9ed0b22be955bb4d481988de9f196fea05ed","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-59c36108e7f99c2f","role":"sequence","name":"basis_map_A013610 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:      3;\n  2:     21,      18;\n  3:    105,     198,       81;\n  4:    465,    1368,     1215,      324;\n  5:   1926,    7722,    10773,     6156,     1215;\n  6:   7650,   38880,    74439,    66744,    27945,     4374;\n  7:  29523,  181818,   443799,   547236,   358425,   118098,   15309;\n  8: 111567,  807372,  2398977,  3781404,  3405645,  1749600,  474579,   52488;\n  9: 414915, 3450474, 12092085, 23262552, 26807760, 18917550, 7975989, 1837080, 177147;\n","recurrence_sentence":"The row polynomials satisfy -(9*n^2-42*n+33-(27*n^2-9*n-18)*x+(27*n-27)*x^2)*P(n,x) = -(72*n^2-237*n+81+(108*n^2-684*n+342)*x-(351*n-459)*x^2+162*x^3)*P(n-1,x) + (39*n-26+(207*n-177)*x+(297*n-405)*x^2+(81*n-351)*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(72*n^2 - 237*n + 81 + (108*n^2 - 684*n + 342)*x - (351*n - 459)*x^2 + 162*x^3)*P[n - 1, x] + (39*n - 26 + (207*n - 177)*x + (297*n - 405)*x^2 + (81*n - 351)*x^3 - 81*x^4)*D[P[n - 1, x], x])/(-9*n^2 + 42*n - 33 + (27*n^2 - 9*n - 18)*x - (27*n - 27)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(72*n^2 - 237*n + 81 + (108*n^2 - 684*n + 342)*x - (351*n - 459)*x^2 + 162*x^3)*v[i - 1] + (39*n - 26 + (207*n - 177)*x + (297*n - 405)*x^2 + (81*n - 351)*x^3 - 81*x^4)*deriv(v[i - 1], x))/(-9*n^2 + 42*n - 33 + (27*n^2 - 9*n - 18)*x - (27*n - 27)*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-d5c14b9fec875ecc","steps":["basis A013610","D"],"edges":["e-16250729b41a60db","e-5d8541f8c1472150"],"text":"A121574 \u2192 basis A013610 \u2192 D"},"headline":"A121574 \u2192 basis A013610 \u2192 D","chain":null}