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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":"f6df0915bfdc0f73a0f643a3ad39224ac0c239328911881bdf27ecb9d08464b1","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:        0;\n  1:        1;\n  2:        7,        2;\n  3:       46,       26,        3;\n  4:      322,      278,       69,        4;\n  5:     2427,     2858,     1134,      164,       5;\n  6:    19645,    29488,    16314,     3964,     375,       6;\n  7:   170236,   311644,   222534,    77044,   12870,     846,      7;\n  8:  1573732,  3407828,  2992938,  1354684,  329690,   40242,   1897,    8;\n  9: 15459637, 38722316, 40476552, 22732516, 7332245, 1335774, 123550, 4232, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-8*n^2+n+(4*n^3-10*n^2+4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (8*n^2-4*n+(12*n^2-8*n)*x+(6*n^2-5*n)*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (4*n^2-6*n+3+(2*n^2-4*n+2)*x)*P(n-1,x) - (4*n-4+(4*n-4)*x+(n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 8*n^2 + n + (4*n^3 - 10*n^2 + 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (8*n^2 - 4*n + (12*n^2 - 8*n)*x + (6*n^2 - 5*n)*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 6*n + 3 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] - (4*n - 4 + (4*n - 4)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 8*n^2 + n + (4*n^3 - 10*n^2 + 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (8*n^2 - 4*n + (12*n^2 - 8*n)*x + (6*n^2 - 5*n)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 6*n + 3 + (2*n^2 - 4*n + 2)*x)*v[i - 1] - (4*n - 4 + (4*n - 4)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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"],"edges":["e-236d0fb15dc6eca0","e-8abcc7766de04335","e-97ce90b02cbe6311"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 x \u21a6 x+1 \u2192 D","chain":null}