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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":"afa4c233eb41c09a2e89d720c6e45d6d26576c97ed332899d939aa7fec116f7a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4689fb1386d440ad","role":"sequence","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  3,    2;\n   3:  7,   18,     6;\n   4: 13,   72,    90,     20;\n   5: 21,  200,   510,    420,      70;\n   6: 31,  450,  1890,   3080,    1890,     252;\n   7: 43,  882,  5460,  14280,   17010,    8316,     924;\n   8: 57, 1568, 13356,  50400,   94710,   88704,   36036,    3432;\n   9: 73, 2592, 28980, 147840,  395010,  576576,  444444,  154440,  12870;\n  10: 91, 4050, 57420, 378840, 1351350, 2774772, 3303300, 2162160, 656370, 48620;\n","recurrence_sentence":"The row polynomials satisfy (n^3-4*n^2+6*n-3+(n-1)*x-(n-1)*x^2)*P(n,x) = (n^3-2*n^2+2*n-1+(2*n^3-4*n^2+3*n-1)*x-(n-1)*x^2-(2*n-2)*x^3)*P(n-1,x) + ((2*n^2-4*n+2)*x+(2*n^2-4*n)*x^2-4*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 2*n^2 + 2*n - 1 + (2*n^3 - 4*n^2 + 3*n - 1)*x - (n - 1)*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + ((2*n^2 - 4*n + 2)*x + (2*n^2 - 4*n)*x^2 - 4*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 6*n - 3 + (n - 1)*x - (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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