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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":"ae45fcddad10dc29cd0942f91f859a9b3c9c54accd0a33d34988347e18d7321e","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-964d853754e5a7e8","role":"sequence","name":"reversal of A133080 * 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:  1;\n  1:  3,  2;\n  2:  3,  4,   1;\n  3:  5, 14,  11,   2;\n  4:  5, 16,  18,   8,    1;\n  5:  7, 34,  62,  52,   19,    2;\n  6:  7, 36,  75,  80,   45,   12,   1;\n  7:  9, 62, 177, 270,  235,  114,  27,   2;\n  8:  9, 64, 196, 336,  350,  224,  84,  16,  1;\n  9: 11, 98, 380, 840, 1162, 1036, 588, 200, 35, 2;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-1+(2*n-2)*x+x^2)*P(n,x) = (3*n^2-n-5+(2*n^3+4*n^2-4*n-4)*x+(3*n^3+n^2+n-2)*x^2+(6*n^2-4*n)*x^3+(3*n-3)*x^4)*P(n-1,x) - (2*n+2+(2*n^2+8*n+6)*x+(5*n^2+13*n+10)*x^2+(3*n^2+13*n+11)*x^3+(6*n+8)*x^4+3*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - n - 5 + (2*n^3 + 4*n^2 - 4*n - 4)*x + (3*n^3 + n^2 + n - 2)*x^2 + (6*n^2 - 4*n)*x^3 + (3*n - 3)*x^4)*P[n - 1, x] - (2*n + 2 + (2*n^2 + 8*n + 6)*x + (5*n^2 + 13*n + 10)*x^2 + (3*n^2 + 13*n + 11)*x^3 + (6*n + 8)*x^4 + 3*x^5)*D[P[n - 1, x], x])/(n^2 - n - 1 + (2*n - 2)*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - n - 5 + (2*n^3 + 4*n^2 - 4*n - 4)*x + (3*n^3 + n^2 + n - 2)*x^2 + (6*n^2 - 4*n)*x^3 + (3*n - 3)*x^4)*v[i - 1] - (2*n + 2 + (2*n^2 + 8*n + 6)*x + (5*n^2 + 13*n + 10)*x^2 + (3*n^2 + 13*n + 11)*x^3 + (6*n + 8)*x^4 + 3*x^5)*deriv(v[i - 1], x))/(n^2 - n - 1 + (2*n - 2)*x + x^2)); 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-d75669a8cd7f9393","steps":["reverse","D+1"],"edges":["e-7d9675c994dd8650","e-22f7f224021fbb44"],"text":"A133087 \u2192 reverse \u2192 D+1"},"headline":"A133087 \u2192 reverse \u2192 D+1","chain":null}