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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":"a61a64dbef82fca504b26bd1dc13a7d6a58ded5d3f2412daae8d65cfdb8e70cf","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-e8e64448d4c6498a","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.","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: 2, 2;\n  2: 1, 7,  3;\n  3: 0, 8, 16,  4;\n  4: 0, 2, 31, 30,   5;\n  5: 0, 0, 18, 86,  50,   6;\n  6: 0, 0,  3, 85, 196,  77,    7;\n  7: 0, 0,  0, 32, 288, 392,  112,    8;\n  8: 0, 0,  0,  4, 181, 792,  714,  156,   9;\n  9: 0, 0,  0,  0,  50, 730, 1884, 1212, 210, 10;\n","recurrence_sentence":"The row polynomials satisfy -(n^3-n^2-(4*n^3-10*n^2+8*n)*x-(8*n^2-14*n)*x^2-4*n*x^3)*P(n,x) = (n^3-4*n^2+3*n-(7*n^3-18*n^2+13*n-6)*x+(12*n^3-34*n^2+30*n-16)*x^2+(24*n^2-42*n+14)*x^3+(12*n-4)*x^4)*P(n-1,x) - ((2*n^2-2*n)*x-(10*n^2-10*n+4)*x^2+(8*n^2-24*n+16)*x^3+(16*n-20)*x^4+8*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[((n^3 - 4*n^2 + 3*n - (7*n^3 - 18*n^2 + 13*n - 6)*x + (12*n^3 - 34*n^2 + 30*n - 16)*x^2 + (24*n^2 - 42*n + 14)*x^3 + (12*n - 4)*x^4)*P[n - 1, x] - ((2*n^2 - 2*n)*x - (10*n^2 - 10*n + 4)*x^2 + (8*n^2 - 24*n + 16)*x^3 + (16*n - 20)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-n^3 + n^2 + (4*n^3 - 10*n^2 + 8*n)*x + (8*n^2 - 14*n)*x^2 + 4*n*x^3)]];\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] = ((n^3 - 4*n^2 + 3*n - (7*n^3 - 18*n^2 + 13*n - 6)*x + (12*n^3 - 34*n^2 + 30*n - 16)*x^2 + (24*n^2 - 42*n + 14)*x^3 + (12*n - 4)*x^4)*v[i - 1] - ((2*n^2 - 2*n)*x - (10*n^2 - 10*n + 4)*x^2 + (8*n^2 - 24*n + 16)*x^3 + (16*n - 20)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-n^3 + n^2 + (4*n^3 - 10*n^2 + 8*n)*x + (8*n^2 - 14*n)*x^2 + 4*n*x^3)); 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-e8e64448d4c6498a","steps":["D+1"],"edges":["e-b9cf8dd811dd60b3"],"text":"A119900 \u2192 D+1"},"headline":"A119900 \u2192 D+1","chain":null}