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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":"dfbdf507d93c302feb8e3925ead3e5018677af8d9b1328c4b9a6e314c43af10e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2a657d152d421a92","role":"sequence","name":"Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.","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:   4,    3;\n   3:   9,   32,     12;\n   4:  16,  125,    220,      55;\n   5:  25,  336,   1248,    1456,     273;\n   6:  36,  735,   4550,   10920,    9520,     1428;\n   7:  49, 1408,  12920,   50592,   89148,    62016,     7752;\n   8:  64, 2457,  31122,  177555,  502740,   697851,   403788,    43263;\n   9:  81, 4000,  66640,  517440, 2107490,  4647104,  5313000,  2631200,   246675;\n  10: 100, 6171, 130548, 1317624, 7225680, 22694100, 40849380, 39665340, 17168580, 1430715;\n","recurrence_sentence":"The row polynomials satisfy (8*n^4-12*n^3+4*n+(4*n^3+10*n^2+4*n)*x-(4*n^2+2*n)*x^2)*P(n,x) = (8*n^4+4*n^3+(36*n^4-26*n^3+16*n^2+4*n)*x+(18*n^3+8*n^2-8*n)*x^2-(18*n^2-6*n)*x^3)*P(n-1,x) + ((16*n^3-18*n^2+10*n-2)*x+(18*n^3-16*n^2+12*n-2)*x^2+(9*n^2-11*n+2)*x^3-(9*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 + 4*n^3 + (36*n^4 - 26*n^3 + 16*n^2 + 4*n)*x + (18*n^3 + 8*n^2 - 8*n)*x^2 - (18*n^2 - 6*n)*x^3)*P[n - 1, x] + ((16*n^3 - 18*n^2 + 10*n - 2)*x + (18*n^3 - 16*n^2 + 12*n - 2)*x^2 + (9*n^2 - 11*n + 2)*x^3 - (9*n - 3)*x^4)*D[P[n - 1, x], x])/(8*n^4 - 12*n^3 + 4*n + (4*n^3 + 10*n^2 + 4*n)*x - (4*n^2 + 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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