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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":"6f40d2f9a6c0af2b59f2883df132aa9543bde4380cfec24635c860912fdfb526","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-212fa067c32397af","role":"sequence","name":"Pascal-(1,6,1) array.","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,    1;\n  2:  9,   10,     1;\n  3: 16,   45,    18,      1;\n  4: 23,  178,   144,     26,      1;\n  5: 30,  409,   760,    306,     34,     1;\n  6: 37,  738,  2895,   2252,    531,    42,     1;\n  7: 44, 1165,  7578,  11695,   5144,   819,    50,    1;\n  8: 51, 1690, 15838,  44070,  34796,  9926,  1170,   58,  1;\n  9: 58, 2313, 28704, 124416, 172836, 83958, 17088, 1584, 66, 1;\n","recurrence_sentence":"The row polynomials satisfy -(7*n^3-19*n^2+12*n-(7*n^3-30*n^2+62*n)*x-(24*n^2-59*n)*x^2-(n^2+22*n)*x^3-n*x^4)*P(n,x) = -(7*n^3-12*n^2+5*n+(91*n^3-320*n^2+151*n)*x+(305*n^2-767*n+493)*x^2+(13*n^2+263*n-287)*x^3+(12*n-37)*x^4-x^5)*P(n-1,x) + ((7*n^2-5*n)*x+(182*n^2-105*n-12)*x^2+(7*n^2+646*n-287)*x^3+(51*n+639)*x^4+(n+51)*x^5+x^6)*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[(-(7*n^3 - 12*n^2 + 5*n + (91*n^3 - 320*n^2 + 151*n)*x + (305*n^2 - 767*n + 493)*x^2 + (13*n^2 + 263*n - 287)*x^3 + (12*n - 37)*x^4 - x^5)*P[n - 1, x] + ((7*n^2 - 5*n)*x + (182*n^2 - 105*n - 12)*x^2 + (7*n^2 + 646*n - 287)*x^3 + (51*n + 639)*x^4 + (n + 51)*x^5 + x^6)*D[P[n - 1, x], x])/(-7*n^3 + 19*n^2 - 12*n + (7*n^3 - 30*n^2 + 62*n)*x + (24*n^2 - 59*n)*x^2 + (n^2 + 22*n)*x^3 + n*x^4)]];\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] = (-(7*n^3 - 12*n^2 + 5*n + (91*n^3 - 320*n^2 + 151*n)*x + (305*n^2 - 767*n + 493)*x^2 + (13*n^2 + 263*n - 287)*x^3 + (12*n - 37)*x^4 - x^5)*v[i - 1] + ((7*n^2 - 5*n)*x + (182*n^2 - 105*n - 12)*x^2 + (7*n^2 + 646*n - 287)*x^3 + (51*n + 639)*x^4 + (n + 51)*x^5 + x^6)*deriv(v[i - 1], x))/(-7*n^3 + 19*n^2 - 12*n + (7*n^3 - 30*n^2 + 62*n)*x + (24*n^2 - 59*n)*x^2 + (n^2 + 22*n)*x^3 + n*x^4)); 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-212fa067c32397af","steps":["D+1"],"edges":["e-e4ee601767287219"],"text":"A081581 \u2192 D+1"},"headline":"A081581 \u2192 D+1","chain":null}