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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":"f9d710afdbda8a924aae0be2a1c208127009157843f35f443c6b8c0dc077b3cc","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-1639e69c5d7e0691","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.","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:        4,        1;\n  2:       18,        9,        1;\n  3:       95,       72,       16,        1;\n  4:      579,      600,      208,       27,        1;\n  5:     3966,     5285,     2563,      542,       46,       1;\n  6:    29944,    49133,    31058,     9402,     1366,      81,      1;\n  7:   246141,   481880,   378356,   151222,    32131,    3420,    148,     1;\n  8:  2183173,  4981560,  4696852,  2352330,   668361,  106180,   8604,   279,   1;\n  9: 20744088, 54181953, 59837710, 36224700, 13015374, 2802223, 346179, 21882, 538, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+3+2*x)*P(n,x) = -(4*n^3+16*n^2-3*n-12+(4*n^3+26*n^2+4*n-22)*x+(n^3+15*n^2+13*n-20)*x^2+(3*n^2+9*n-10)*x^3+(2*n-2)*x^4)*P(n-2,x) + (8*n^2+44*n+48+(12*n^2+88*n+120)*x+(6*n^2+67*n+124)*x^2+(n^2+23*n+66)*x^3+(3*n+18)*x^4+2*x^5)*d/dx P(n-2,x) + (4*n^2+14*n+3+(2*n^2+16*n+6)*x+(5*n+6)*x^2+2*x^3)*P(n-1,x) - (4*n+12+(4*n+20)*x+(n+11)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 + 16*n^2 - 3*n - 12 + (4*n^3 + 26*n^2 + 4*n - 22)*x + (n^3 + 15*n^2 + 13*n - 20)*x^2 + (3*n^2 + 9*n - 10)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + (8*n^2 + 44*n + 48 + (12*n^2 + 88*n + 120)*x + (6*n^2 + 67*n + 124)*x^2 + (n^2 + 23*n + 66)*x^3 + (3*n + 18)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (4*n^2 + 14*n + 3 + (2*n^2 + 16*n + 6)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] - (4*n + 12 + (4*n + 20)*x + (n + 11)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 3 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 + 16*n^2 - 3*n - 12 + (4*n^3 + 26*n^2 + 4*n - 22)*x + (n^3 + 15*n^2 + 13*n - 20)*x^2 + (3*n^2 + 9*n - 10)*x^3 + (2*n - 2)*x^4)*v[i - 2] + (8*n^2 + 44*n + 48 + (12*n^2 + 88*n + 120)*x + (6*n^2 + 67*n + 124)*x^2 + (n^2 + 23*n + 66)*x^3 + (3*n + 18)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (4*n^2 + 14*n + 3 + (2*n^2 + 16*n + 6)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] - (4*n + 12 + (4*n + 20)*x + (n + 11)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 3 + 2*x)); 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-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","x \u21a6 x+1","D+1"],"edges":["e-236d0fb15dc6eca0","e-8abcc7766de04335","e-d83787b6015d12f1"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}