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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":"5f2b384ef7da8706639cf33b6b6e35e4f6857ad29c2ba634f6c96359176bdb00","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-15cf3f3457596bec","role":"sequence","name":"veronese_section_2_1 of Triangle of coefficients in expansion of (1+3*x)^n.","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:    6;\n  3:   36,    27;\n  4:  120,   108;\n  5:  285,   756,    243;\n  6:  558,  3456,   1458;\n  7:  966, 11151,  11664,   2187;\n  8: 1536, 28728,  66096,  17496;\n  9: 2295, 63504, 266814, 157464, 19683;\n","recurrence_sentence":"The row polynomials satisfy (9*n^3-54*n^2+105*n-60+(36*n^2-166*n+130)*x+(36*n-36)*x^2)*P(n,x) = (9*n^3-27*n^2+24*n+(81*n^3-531*n^2+1022*n-684)*x+(324*n^2-1458*n+1692)*x^2+(324*n-648)*x^3)*P(n-1,x) + ((18*n^2-54*n+40)*x-(162*n^2-558*n+476)*x^2-(648*n-1116)*x^3-648*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[((9*n^3 - 27*n^2 + 24*n + (81*n^3 - 531*n^2 + 1022*n - 684)*x + (324*n^2 - 1458*n + 1692)*x^2 + (324*n - 648)*x^3)*P[n - 1, x] + ((18*n^2 - 54*n + 40)*x - (162*n^2 - 558*n + 476)*x^2 - (648*n - 1116)*x^3 - 648*x^4)*D[P[n - 1, x], x])/(9*n^3 - 54*n^2 + 105*n - 60 + (36*n^2 - 166*n + 130)*x + (36*n - 36)*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] = ((9*n^3 - 27*n^2 + 24*n + (81*n^3 - 531*n^2 + 1022*n - 684)*x + (324*n^2 - 1458*n + 1692)*x^2 + (324*n - 648)*x^3)*v[i - 1] + ((18*n^2 - 54*n + 40)*x - (162*n^2 - 558*n + 476)*x^2 - (648*n - 1116)*x^3 - 648*x^4)*deriv(v[i - 1], x))/(9*n^3 - 54*n^2 + 105*n - 60 + (36*n^2 - 166*n + 130)*x + (36*n - 36)*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-6a5af5e1168bb679","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-a72a0a226c85ec5c","e-8fdb1695ef1ae7cb"],"text":"A013610 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A013610 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}