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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":"d19dbe6a1a9f04bfc1fea50896c0ccd694a2f16749f105b1cef5e361602a913b","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-36098d4730252f73","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.","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:      2,       1;\n   3:      8,       8,       1;\n   4:     35,      60,      18,       1;\n   5:    154,     420,     224,      32,       1;\n   6:    672,    2730,    2310,     600,      50,       1;\n   7:   2904,   16632,   20790,    8580,    1320,      72,      1;\n   8:  12441,   96096,  168168,  102102,   25025,    2548,     98,    1;\n   9:  52910,  531960, 1249248, 1057056,  380380,   61880,   4480,  128,   1;\n  10: 223652, 2844270, 8664084, 9801792, 4900896, 1169532, 135660, 7344, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^4-7*n^3+11*n^2-4*n+(4*n^3-18*n^2+8*n)*x+(2*n^2-n)*x^2)*P(n,x) = (8*n^4-24*n^3+38*n^2-38*n+12+(6*n^4-10*n^3-72*n^2+148*n-60)*x+(12*n^3-58*n^2+80*n-36)*x^2+(6*n^2-12*n+6)*x^3)*P(n-1,x) + ((16*n^3-36*n^2-12*n+24)*x-(4*n^3-41*n^2+117*n-66)*x^2-(8*n^2-46*n+30)*x^3-(4*n-3)*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[((8*n^4 - 24*n^3 + 38*n^2 - 38*n + 12 + (6*n^4 - 10*n^3 - 72*n^2 + 148*n - 60)*x + (12*n^3 - 58*n^2 + 80*n - 36)*x^2 + (6*n^2 - 12*n + 6)*x^3)*P[n - 1, x] + ((16*n^3 - 36*n^2 - 12*n + 24)*x - (4*n^3 - 41*n^2 + 117*n - 66)*x^2 - (8*n^2 - 46*n + 30)*x^3 - (4*n - 3)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 7*n^3 + 11*n^2 - 4*n + (4*n^3 - 18*n^2 + 8*n)*x + (2*n^2 - n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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