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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":"b7c91b34f37d16d361393f5c43b187c277de709905bbdc2d8217e349e2225978","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-b4814297dd192d81","role":"sequence","name":"Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.","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":[{"id":"l-ac01801868d5dff1","kind":"oeis_match","node_a_id":"n-ade3ecc7e8430e63","node_b_id":null,"fingerprint":null,"oeis_id":"A034428","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A034428","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A034428","relation":"match_row_sums"}],"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:   1;\n  2:   2,     1;\n  3:   5,     4;\n  4:  12,    19,     4;\n  5:  27,    94,    34;\n  6:  58,   417,   282,    34;\n  7: 121,  1656,  2256,   496;\n  8: 248,  6055, 15768,  6272,   496;\n  9: 503, 20890, 96498, 72992, 11056;\n","recurrence_sentence":"The row polynomials satisfy (n-3+4*x)*P(n,x) = -(2*n-4+(2*n^2-13*n+26)*x+(n^3-9*n^2+34*n-60)*x^2+(6*n^2-36*n+56)*x^3+(8*n-16)*x^4)*P(n-2,x) - ((n-2)*x+(n^2-7*n+14)*x^2-(2*n^2-16*n+36)*x^3-(12*n-40)*x^4-16*x^5)*d/dx P(n-2,x) + (3*n-8+(2*n^2-13*n+33)*x+(10*n-36)*x^2+8*x^3)*P(n-1,x) + ((n-3)*x-(2*n-10)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (2*n^2 - 13*n + 26)*x + (n^3 - 9*n^2 + 34*n - 60)*x^2 + (6*n^2 - 36*n + 56)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] - ((n - 2)*x + (n^2 - 7*n + 14)*x^2 - (2*n^2 - 16*n + 36)*x^3 - (12*n - 40)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (3*n - 8 + (2*n^2 - 13*n + 33)*x + (10*n - 36)*x^2 + 8*x^3)*P[n - 1, x] + ((n - 3)*x - (2*n - 10)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(n - 3 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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