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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":"4f754e677cc7659ee44d4eee116ac8530ce40a0ef00af31c16ac9d6024897209","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-e6d483569f3966cd","role":"sequence","name":"basis_map_A013610 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:       6;\n  2:      54,       72;\n  3:     360,     1008,       648;\n  4:    2136,     9144,     12312,       5184;\n  5:   11880,    68544,    141912,     124416,      38880;\n  6:   63432,   460656,   1291464,    1741824,    1127520,     279936;\n  7:  329280,  2883456,  10212480,   18683136,   18584640,    9517824,    1959552;\n  8: 1674432, 17167680,  73529856,  170247744,  229780800,  180558720,   76422528,  13436928;\n  9: 8381664, 98482176, 495004608, 1388295936, 2373274080, 2529501696, 1640145024, 591224832, 90699264;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2+(3*n^2-3*n)*x-(6*n-6)*x^2)*P(n,x) = -(6*n^2-28*n+14+(12*n^2-102*n+78)*x-(78*n-138)*x^2+72*x^3)*P(n-1,x) + (4*n-4+(24*n-32)*x+(42*n-90)*x^2+(18*n-102)*x^3-36*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 - 28*n + 14 + (12*n^2 - 102*n + 78)*x - (78*n - 138)*x^2 + 72*x^3)*P[n - 1, x] + (4*n - 4 + (24*n - 32)*x + (42*n - 90)*x^2 + (18*n - 102)*x^3 - 36*x^4)*D[P[n - 1, x], x])/(2*n - 2 + (3*n^2 - 3*n)*x - (6*n - 6)*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] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n^2 - 28*n + 14 + (12*n^2 - 102*n + 78)*x - (78*n - 138)*x^2 + 72*x^3)*v[i - 1] + (4*n - 4 + (24*n - 32)*x + (42*n - 90)*x^2 + (18*n - 102)*x^3 - 36*x^4)*deriv(v[i - 1], x))/(2*n - 2 + (3*n^2 - 3*n)*x - (6*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-1bffaa6198344f93","steps":["basis A013610","D"],"edges":["e-04dfb33ef5bb2a04","e-b557037e14dfa313"],"text":"A059473 \u2192 basis A013610 \u2192 D"},"headline":"A059473 \u2192 basis A013610 \u2192 D","chain":null}