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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":"8f2900d421a2a789ab8fa159016af3eb81b8687cd9534a8af2129a4694948550","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-8e6b51ac707373f6","role":"sequence","name":"Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.","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:       8,        9,        2;\n  3:      40,       70,       36,        5;\n  4:     220,      525,      432,      140,       14;\n  5:    1274,     3861,     4400,     2310,      540,       42;\n  6:    7616,    28028,    40950,    30030,    11440,     2079,     132;\n  7:   46512,   201552,   359856,   340340,   182000,    54054,    8008,     429;\n  8:  288420,  1438965,  3038784,  3527160,  2441880,  1021020,  247520,   30888,   1430;\n  9: 1808950, 10214875, 24919488, 34321980, 29248296, 15872220, 5426400, 1108536, 119340, 4862;\n","recurrence_sentence":"The row polynomials satisfy (8*n^4+4*n^3-8*n^2-4*n+(12*n^3-2*n^2-4*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (54*n^4-150*n^2+120*n-24+(48*n^4+73*n^3-232*n^2+153*n-30)*x+(2*n^4+68*n^3-67*n^2+17*n-2)*x^2+(3*n^3+18*n^2-20*n+5)*x^3+(n^2-2*n+1)*x^4)*P(n-1,x) - ((18*n^3+18*n^2-32*n+8)*x+(20*n^3+45*n^2-45*n+10)*x^2+(2*n^3+30*n^2-6*n-2)*x^3+(3*n^2+8*n-5)*x^4+(n-1)*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[((54*n^4 - 150*n^2 + 120*n - 24 + (48*n^4 + 73*n^3 - 232*n^2 + 153*n - 30)*x + (2*n^4 + 68*n^3 - 67*n^2 + 17*n - 2)*x^2 + (3*n^3 + 18*n^2 - 20*n + 5)*x^3 + (n^2 - 2*n + 1)*x^4)*P[n - 1, x] - ((18*n^3 + 18*n^2 - 32*n + 8)*x + (20*n^3 + 45*n^2 - 45*n + 10)*x^2 + (2*n^3 + 30*n^2 - 6*n - 2)*x^3 + (3*n^2 + 8*n - 5)*x^4 + (n - 1)*x^5)*D[P[n - 1, x], x])/(8*n^4 + 4*n^3 - 8*n^2 - 4*n + (12*n^3 - 2*n^2 - 4*n)*x + (4*n^2 + 2*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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