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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":"a1164972e81b84ad2203a8ad737b31d72f610e3bb80557a1e5c23190f61d5b05","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-f339d17ab0e2d2a9","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","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:       2;\n  2:      13,        8;\n  3:      71,       86,       24;\n  4:     383,      692,      387,       64;\n  5:    2129,     5132,     4347,     1484,      160;\n  6:   12360,    37382,    42732,    22544,     5245,     384;\n  7:   75242,   274486,   397002,   286444,   105555,   17730,     896;\n  8:  480513,  2056784,  3610779,  3326684,  1701625,  465828,   58471,   2048;\n  9: 3216113, 15825164, 32755251, 36874156, 24322615, 9393096, 1984115, 190232, 4608;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+2*n^2-1+(2*n^3+2*n^2-2*n-2)*x+(n^3-n)*x^2)*P(n-2,x) + (n^2+2*n+1+(3*n^2+5*n+2)*x+(3*n^2+4*n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n^2+2*n+1+2*n^2*x)*P(n-1,x) - (n+2*n*x+n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 2*n^2 - 1 + (2*n^3 + 2*n^2 - 2*n - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] + (n^2 + 2*n + 1 + (3*n^2 + 5*n + 2)*x + (3*n^2 + 4*n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 2*n + 1 + 2*n^2*x)*P[n - 1, x] - (n + 2*n*x + n*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 2*n^2 - 1 + (2*n^3 + 2*n^2 - 2*n - 2)*x + (n^3 - n)*x^2)*v[i - 2] + (n^2 + 2*n + 1 + (3*n^2 + 5*n + 2)*x + (3*n^2 + 4*n + 1)*x^2 + (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 2*n + 1 + 2*n^2*x)*v[i - 1] - (n + 2*n*x + n*x^2)*deriv(v[i - 1], x))/(n)); 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-0990b5be9f344750","steps":["x \u21a6 x+1","D"],"edges":["e-2b15edbdb8f7710e","e-b85dd30f9f722397"],"text":"A126351 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A126351 \u2192 x \u21a6 x+1 \u2192 D","chain":null}