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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":"38642242b0056c2988fcf9179d68eaaa52e33a82005ee73d68a4103d2cf4f7f3","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-8f767a3615460a6b","role":"sequence","name":"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:      5,       2;\n  3:     23,      20,       3;\n  4:    109,     152,      57,       4;\n  5:    544,    1062,     672,     144,       5;\n  6:   2876,    7282,    6612,    2524,     345,      6;\n  7:  16113,   50416,   60267,   33884,    8745,    804,     7;\n  8:  95495,  357128,  532893,  400684,  156655,  29028,  1841,    8;\n  9: 597155, 2604902, 4679364, 4437892, 2359135, 682842, 94178, 4160, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3-n^2-n+(2*n^3-4*n^2)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (n^2+(3*n^2-n)*x+(3*n^2-2*n)*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2-2*n+1+(2*n^2-4*n+2)*x)*P(n-1,x) - (n-1+(2*n-2)*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[(-(n^3 - n^2 - n + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (n^2 + (3*n^2 - n)*x + (3*n^2 - 2*n)*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] - (n - 1 + (2*n - 2)*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] = (-(n^3 - n^2 - n + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (n^2 + (3*n^2 - n)*x + (3*n^2 - 2*n)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x)*v[i - 1] - (n - 1 + (2*n - 2)*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)","D"],"edges":["e-236d0fb15dc6eca0","e-1c9f283a75acfd0f"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 D"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 D","chain":null}