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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":"5fe64d9954eabcddadc68b0b3186bc339659afebc27f21b4993c9f32c638a425","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-2e7e4df17772fe4d","role":"sequence","name":"reversal of Triangle read by rows: A008277 * A007318.","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:  5,   10;\n  3:  9,   44,    45;\n  4: 14,  122,   297,    208;\n  5: 20,  270,  1155,   1884,    1015;\n  6: 27,  520,  3420,   9664,   11930,    5262;\n  7: 35,  910,  8505,  36624,   77350,   77202,   28980;\n  8: 44, 1484, 18690, 113764,  362950,  613290,  515767,  169176;\n  9: 54, 2292, 37422, 306012, 1371930, 3473574, 4902177, 3572960, 1043775;\n","recurrence_sentence":"The row polynomials satisfy (n+(2*n-4)*x)*P(n,x) = -(n+1+(2*n^2+2*n-4)*x+(n^3+6*n^2-14*n-3)*x^2+(4*n^3-4*n^2-22*n+14)*x^3+(5*n^3-20*n^2+13*n+6)*x^4+(2*n^3-12*n^2+22*n-12)*x^5)*P(n-2,x) + ((n+1)*x^2+(n^2+4*n-1)*x^3+(4*n^2-4)*x^4+(5*n^2-9*n+2)*x^5+(2*n^2-6*n+4)*x^6)*d/dx P(n-2,x) + (2*n+1+(2*n^2+4*n-6)*x+(6*n^2-13*n+2)*x^2+(4*n^2-18*n+20)*x^3)*P(n-1,x) - (n*x^2+(3*n-4)*x^3+(2*n-4)*x^4)*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 + 1 + (2*n^2 + 2*n - 4)*x + (n^3 + 6*n^2 - 14*n - 3)*x^2 + (4*n^3 - 4*n^2 - 22*n + 14)*x^3 + (5*n^3 - 20*n^2 + 13*n + 6)*x^4 + (2*n^3 - 12*n^2 + 22*n - 12)*x^5)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + 4*n - 1)*x^3 + (4*n^2 - 4)*x^4 + (5*n^2 - 9*n + 2)*x^5 + (2*n^2 - 6*n + 4)*x^6)*D[P[n - 2, x], x] + (2*n + 1 + (2*n^2 + 4*n - 6)*x + (6*n^2 - 13*n + 2)*x^2 + (4*n^2 - 18*n + 20)*x^3)*P[n - 1, x] - (n*x^2 + (3*n - 4)*x^3 + (2*n - 4)*x^4)*D[P[n - 1, x], x])/(n + (2*n - 4)*x)]];\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 + 1 + (2*n^2 + 2*n - 4)*x + (n^3 + 6*n^2 - 14*n - 3)*x^2 + (4*n^3 - 4*n^2 - 22*n + 14)*x^3 + (5*n^3 - 20*n^2 + 13*n + 6)*x^4 + (2*n^3 - 12*n^2 + 22*n - 12)*x^5)*v[i - 2] + ((n + 1)*x^2 + (n^2 + 4*n - 1)*x^3 + (4*n^2 - 4)*x^4 + (5*n^2 - 9*n + 2)*x^5 + (2*n^2 - 6*n + 4)*x^6)*deriv(v[i - 2], x) + (2*n + 1 + (2*n^2 + 4*n - 6)*x + (6*n^2 - 13*n + 2)*x^2 + (4*n^2 - 18*n + 20)*x^3)*v[i - 1] - (n*x^2 + (3*n - 4)*x^3 + (2*n - 4)*x^4)*deriv(v[i - 1], x))/(n + (2*n - 4)*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-bdae5aa3d1cac0ec","steps":["reverse","D"],"edges":["e-475fdd3e596ddb02","e-9f583f90e860787a"],"text":"A137597 \u2192 reverse \u2192 D"},"headline":"A137597 \u2192 reverse \u2192 D","chain":null}