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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":"1867b3a536645ee53a14c6b2d03595ae42e417250363eae36af85ed01726b27f","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-e8d6574c8023ca59","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-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:       1;\n  2:       5,       2;\n  3:      17,       8;\n  4:      93,      86,      16;\n  5:     513,     596,     136;\n  6:    3477,    5442,    2352,     272;\n  7:   25569,   48240,   26880,    3968;\n  8:  212733,  482670,  360288,   97792,   7936;\n  9: 1929393, 5065740, 4605768, 1645312, 176896;\n","recurrence_sentence":"The row polynomials satisfy (2*n+3+8*x)*P(n,x) = -(8*n^3-8*n^2-52*n+45+(16*n^3+24*n^2-202*n+147)*x+(8*n^3+80*n^2-218*n+80)*x^2+(48*n^2-8*n-144)*x^3+(64*n-128)*x^4)*P(n-2,x) + (12*n^2+30*n+(40*n^2+166*n+45)*x+(44*n^2+336*n+237)*x^2+(16*n^2+296*n+464)*x^3+(96*n+400)*x^4+128*x^5)*d/dx P(n-2,x) + (8*n^2-24+(8*n^2+34*n-79)*x+(40*n-36)*x^2+32*x^3)*P(n-1,x) - (6*n+9+(14*n+45)*x+(8*n+68)*x^2+32*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 8*n^2 - 52*n + 45 + (16*n^3 + 24*n^2 - 202*n + 147)*x + (8*n^3 + 80*n^2 - 218*n + 80)*x^2 + (48*n^2 - 8*n - 144)*x^3 + (64*n - 128)*x^4)*P[n - 2, x] + (12*n^2 + 30*n + (40*n^2 + 166*n + 45)*x + (44*n^2 + 336*n + 237)*x^2 + (16*n^2 + 296*n + 464)*x^3 + (96*n + 400)*x^4 + 128*x^5)*D[P[n - 2, x], x] + (8*n^2 - 24 + (8*n^2 + 34*n - 79)*x + (40*n - 36)*x^2 + 32*x^3)*P[n - 1, x] - (6*n + 9 + (14*n + 45)*x + (8*n + 68)*x^2 + 32*x^3)*D[P[n - 1, x], x])/(2*n + 3 + 8*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 8*n^2 - 52*n + 45 + (16*n^3 + 24*n^2 - 202*n + 147)*x + (8*n^3 + 80*n^2 - 218*n + 80)*x^2 + (48*n^2 - 8*n - 144)*x^3 + (64*n - 128)*x^4)*v[i - 2] + (12*n^2 + 30*n + (40*n^2 + 166*n + 45)*x + (44*n^2 + 336*n + 237)*x^2 + (16*n^2 + 296*n + 464)*x^3 + (96*n + 400)*x^4 + 128*x^5)*deriv(v[i - 2], x) + (8*n^2 - 24 + (8*n^2 + 34*n - 79)*x + (40*n - 36)*x^2 + 32*x^3)*v[i - 1] - (6*n + 9 + (14*n + 45)*x + (8*n + 68)*x^2 + 32*x^3)*deriv(v[i - 1], x))/(2*n + 3 + 8*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-ff270bcb802a85af","steps":["\u03b3 (shift +1)","x \u21a6 x+1","D+1"],"edges":["e-7e5061201b6d9b87","e-33f9ec87155c6d6e","e-9b4a952557ebeaa2"],"text":"A008292 \u2192 \u03b3 (shift +1) \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A008292 \u2192 \u03b3 (shift +1) \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}