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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":"348e8e2a5a927bfaaf79311c31c88e3eb10fa4f752f72a10b90638a7d7f1b47c","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-43836771acfe9d89","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:       6,        2;\n  2:      36,       26,        4;\n  3:     200,      232,       80,        8;\n  4:    1068,     1728,      964,      216,       16;\n  5:    5544,    11576,     9064,     3288,      544,      32;\n  6:   28192,    72328,    73424,    37448,    10016,    1312,      64;\n  7:  141120,   430144,   538432,   356736,   134144,   28352,    3072,    128;\n  8:  697680,  2465664,  3677088,  3009600,  1471056,  436800,   76224,   7040,   256;\n  9: 3414752, 13736352, 23804736, 23250624, 14051424, 5423136, 1328256, 197248, 15872, 512;\n","recurrence_sentence":"The row polynomials satisfy (6*n^2+6*n+(3*n^3+9*n^2+6*n)*x+(10*n^2+4*n)*x^2+(2*n^2+8*n)*x^3+2*n*x^4)*P(n,x) = -(18*n^3-12*n^2-90*n+12+(12*n^3+18*n^2-138*n-24)*x+(34*n^2-42*n-112)*x^2+(8*n^2+12*n-108)*x^3+(4*n-36)*x^4-4*x^5)*P(n-1,x) + (36*n^2+108*n+72+(72*n^2+336*n+336)*x+(42*n^2+390*n+612)*x^2+(6*n^2+206*n+548)*x^3+(48*n+248)*x^4+(4*n+52)*x^5+4*x^6)*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[(-(18*n^3 - 12*n^2 - 90*n + 12 + (12*n^3 + 18*n^2 - 138*n - 24)*x + (34*n^2 - 42*n - 112)*x^2 + (8*n^2 + 12*n - 108)*x^3 + (4*n - 36)*x^4 - 4*x^5)*P[n - 1, x] + (36*n^2 + 108*n + 72 + (72*n^2 + 336*n + 336)*x + (42*n^2 + 390*n + 612)*x^2 + (6*n^2 + 206*n + 548)*x^3 + (48*n + 248)*x^4 + (4*n + 52)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(6*n^2 + 6*n + (3*n^3 + 9*n^2 + 6*n)*x + (10*n^2 + 4*n)*x^2 + (2*n^2 + 8*n)*x^3 + 2*n*x^4)]];\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] = (-(18*n^3 - 12*n^2 - 90*n + 12 + (12*n^3 + 18*n^2 - 138*n - 24)*x + (34*n^2 - 42*n - 112)*x^2 + (8*n^2 + 12*n - 108)*x^3 + (4*n - 36)*x^4 - 4*x^5)*v[i - 1] + (36*n^2 + 108*n + 72 + (72*n^2 + 336*n + 336)*x + (42*n^2 + 390*n + 612)*x^2 + (6*n^2 + 206*n + 548)*x^3 + (48*n + 248)*x^4 + (4*n + 52)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(6*n^2 + 6*n + (3*n^3 + 9*n^2 + 6*n)*x + (10*n^2 + 4*n)*x^2 + (2*n^2 + 8*n)*x^3 + 2*n*x^4)); 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-1bffaa6198344f93","steps":["x \u21a6 x+1","D+1"],"edges":["e-c090df23694a462d","e-10c0b3c9dc639bf7"],"text":"A059473 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A059473 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}