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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":"5c4846607d057b945668dfa6e38ffe41951999a89d49e5cb8b366b198d09ad67","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-e189038e95d51606","role":"sequence","name":"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:  4,   3;\n  3:  7,   6;\n  4: 10,  27,    9;\n  5: 13,  66,   27;\n  6: 16, 123,  135,   27;\n  7: 19, 198,  414,  108;\n  8: 22, 291,  945,  594,  81;\n  9: 25, 402, 1809, 2160, 405;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-8*n^2+5*n+(12*n^2-29*n)*x+12*n*x^2)*P(n,x) = (3*n^3-5*n^2+2*n+(18*n^3-54*n^2+19*n)*x+(72*n^2-162*n+102)*x^2+(72*n-72)*x^3)*P(n-1,x) - ((3*n^2-2*n)*x+(36*n^2-12*n-5)*x^2+(144*n-48)*x^3+144*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[((3*n^3 - 5*n^2 + 2*n + (18*n^3 - 54*n^2 + 19*n)*x + (72*n^2 - 162*n + 102)*x^2 + (72*n - 72)*x^3)*P[n - 1, x] - ((3*n^2 - 2*n)*x + (36*n^2 - 12*n - 5)*x^2 + (144*n - 48)*x^3 + 144*x^4)*D[P[n - 1, x], x])/(3*n^3 - 8*n^2 + 5*n + (12*n^2 - 29*n)*x + 12*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] = ((3*n^3 - 5*n^2 + 2*n + (18*n^3 - 54*n^2 + 19*n)*x + (72*n^2 - 162*n + 102)*x^2 + (72*n - 72)*x^3)*v[i - 1] - ((3*n^2 - 2*n)*x + (36*n^2 - 12*n - 5)*x^2 + (144*n - 48)*x^3 + 144*x^4)*deriv(v[i - 1], x))/(3*n^3 - 8*n^2 + 5*n + (12*n^2 - 29*n)*x + 12*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":["D+1"],"edges":["e-8635ad6e77f468ea"],"text":"A304236 \u2192 D+1"},"headline":"A304236 \u2192 D+1","chain":null}