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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":"0b1d5a7b57d0fa84bbb899ee28a0459f1c912abcccd28e06cb7e664f01db8e08","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-8d5fed5e270db201","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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:     1;\n  1:     1;\n  2:    13,     9;\n  3:    25,    18;\n  4:   100,   243,     81;\n  5:   238,   684,    243;\n  6:   709,  3042,   3402,    729;\n  7:  1783,  9018,  12474,   2916;\n  8:  4783, 30033,  60993,  42282,  6561;\n  9: 12085, 87228, 213111, 189540, 32805;\n","recurrence_sentence":"The row polynomials satisfy (81*n^3-126*n^2-78*n+(324*n^2-585*n)*x+324*n*x^2)*P(n,x) = (567*n^3-1287*n^2-141*n+738+(1458*n^3-1458*n^2-4041*n+3780)*x+(5832*n^2-8262*n+2754)*x^2+(5832*n-5832)*x^3)*P(n-1,x) - (351*n^2+156*n-65+(2025*n^2+2304*n-102)*x+(2916*n^2+9396*n+2439)*x^2+(11664*n+10368)*x^3+11664*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((567*n^3 - 1287*n^2 - 141*n + 738 + (1458*n^3 - 1458*n^2 - 4041*n + 3780)*x + (5832*n^2 - 8262*n + 2754)*x^2 + (5832*n - 5832)*x^3)*P[n - 1, x] - (351*n^2 + 156*n - 65 + (2025*n^2 + 2304*n - 102)*x + (2916*n^2 + 9396*n + 2439)*x^2 + (11664*n + 10368)*x^3 + 11664*x^4)*D[P[n - 1, x], x])/(81*n^3 - 126*n^2 - 78*n + (324*n^2 - 585*n)*x + 324*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((567*n^3 - 1287*n^2 - 141*n + 738 + (1458*n^3 - 1458*n^2 - 4041*n + 3780)*x + (5832*n^2 - 8262*n + 2754)*x^2 + (5832*n - 5832)*x^3)*v[i - 1] - (351*n^2 + 156*n - 65 + (2025*n^2 + 2304*n - 102)*x + (2916*n^2 + 9396*n + 2439)*x^2 + (11664*n + 10368)*x^3 + 11664*x^4)*deriv(v[i - 1], x))/(81*n^3 - 126*n^2 - 78*n + (324*n^2 - 585*n)*x + 324*n*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-e189038e95d51606","steps":["basis A013610","D+1"],"edges":["e-99e9156f8de29382","e-a93b3259037121a5"],"text":"A304236 \u2192 basis A013610 \u2192 D+1"},"headline":"A304236 \u2192 basis A013610 \u2192 D+1","chain":null}