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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":"9581ba67f4b288da90d3e8048b8a245bdd60a781502d2e3f3f4b2a4a78612d58","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-be392fc9b98fb771","role":"sequence","name":"Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..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":[{"id":"l-53f36b69d3dec681","kind":"fingerprint_related","node_a_id":"n-cc0179e01d57c43d","node_b_id":"n-854b531da9a10af4","fingerprint":"f6-695598672c6e8bd2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-695598672c6e8bd2"},"other_id":"n-854b531da9a10af4","other":{"id":"n-854b531da9a10af4","name":"one_plus_derivative of Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-695598672c6e8bd2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","42","12"]],"card_rows":[["1"],["4","2"],["24","42","12"],["168","624","600","120"]],"row_sums_preview":["1","6","78","1512","38520","1211760","45304560","1961124480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-694017a14c5c1f2e","steps":["D+1"],"edges":["e-b2374e2588a9a86b"],"text":"A253284 \u2192 D+1"},"headline":"A253284 \u2192 D+1","chain":null}}],"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:  4,    7,     2;\n   4:  7,   26,    25,     5;\n   5: 11,   70,   135,    91,     14;\n   6: 16,  155,   490,   644,    336,     42;\n   7: 22,  301,  1400,  2940,   2940,   1254,    132;\n   8: 29,  532,  3402, 10290,  16170,  13068,   4719,    429;\n   9: 37,  876,  7350, 30030,  66990,  84084,  57057,  17875,  1430;\n  10: 46, 1365, 14520, 76692, 228228, 402402, 420420, 245960, 68068, 4862;\n","recurrence_sentence":"The row polynomials satisfy (n^3-3*n^2+4*n+n*x-n*x^2)*P(n,x) = (n^3-n^2+2*n+(2*n^3-4*n^2+5*n-2)*x-n*x^2-(2*n-2)*x^3)*P(n-1,x) + ((2*n^2-4*n+4)*x+(2*n^2-4*n+2)*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 - n^2 + 2*n + (2*n^3 - 4*n^2 + 5*n - 2)*x - n*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + ((2*n^2 - 4*n + 4)*x + (2*n^2 - 4*n + 2)*x^2 - 4*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 4*n + n*x - n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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