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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":"5869929b284f6fd08684e8bf094c69adb31ab3f6d0c5c84e7db8913656a1caa2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c63944481deaae45","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,      1;\n   3:    11,      7,      1;\n   4:    36,     36,     11,      1;\n   5:   116,    158,     76,     15,     1;\n   6:   364,    632,    419,    131,    19,     1;\n   7:  1124,   2376,   2017,    873,   201,    23,    1;\n   8:  3424,   8544,   8864,   4928,  1574,   286,   27,   1;\n   9: 10320,  29712,  36480,  24844, 10211,  2576,  386,  31,  1;\n  10: 30832, 100656, 142872, 115360, 58167, 18901, 3933, 501, 35, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3+6*n^2-13*n+6+(7*n^2-3*n-4)*x+(n^2+5*n-6)*x^2+(n-1)*x^3)*P(n,x) = -(2*n^3-26*n+(11*n^2-31*n-48)*x+(2*n^2-50)*x^2+(n-13)*x^3-x^4)*P(n-1,x) + (12*n^2+84*n+96+(8*n^2+140*n+244)*x+(n^2+75*n+224)*x^2+(15*n+91)*x^3+(n+16)*x^4+x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 26*n + (11*n^2 - 31*n - 48)*x + (2*n^2 - 50)*x^2 + (n - 13)*x^3 - x^4)*P[n - 1, x] + (12*n^2 + 84*n + 96 + (8*n^2 + 140*n + 244)*x + (n^2 + 75*n + 224)*x^2 + (15*n + 91)*x^3 + (n + 16)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 + 6*n^2 - 13*n + 6 + (7*n^2 - 3*n - 4)*x + (n^2 + 5*n - 6)*x^2 + (n - 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 26*n + (11*n^2 - 31*n - 48)*x + (2*n^2 - 50)*x^2 + (n - 13)*x^3 - x^4)*v[i - 1] + (12*n^2 + 84*n + 96 + (8*n^2 + 140*n + 244)*x + (n^2 + 75*n + 224)*x^2 + (15*n + 91)*x^3 + (n + 16)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 + 6*n^2 - 13*n + 6 + (7*n^2 - 3*n - 4)*x + (n^2 + 5*n - 6)*x^2 + (n - 1)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["basis A007318","D+1"],"edges":["e-3a20d12bade77e7f","e-e73282ef15bfa15f"],"text":"A063967 \u2192 basis A007318 \u2192 D+1"},"headline":"A063967 \u2192 basis A007318 \u2192 D+1","chain":null}