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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":"ffd189eb30747af0f799a80caeb707e0ad89740d30985d7c8ba40f1a1251b92f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7c95902cf1293962","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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":[],"oeis":[],"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:           0;\n  1:           4;\n  2:          28,          32;\n  3:         264,         480,         192;\n  4:        3224,        7392,        4992,        1024;\n  5:       48656,      129280,      114240,       40960,        5120;\n  6:      876432,     2587008,     2717760,     1305600,      291840,      24576;\n  7:    18366400,    58751616,    69877248,    40606720,    12364800,    1892352,    114688;\n  8:   439182400,  1498218368,  1959686400,  1306633216,   487424000,  102875136,  11468800,   524288;\n  9: 11804112000, 42467136768, 59941156608, 44424124416, 19206190080, 5014388736, 778272768, 66060288, 2359296;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(24*n-32)*x+16*x^2)*P(n-2,x) + (6*n-5+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (24*n - 32)*x + 16*x^2)*P[n - 2, x] + (6*n - 5 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (24*n - 32)*x + 16*x^2)*v[i - 2] + (6*n - 5 + 8*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-260e022530e5976c","steps":["basis A013611","D"],"edges":["e-10584eb473514fc3","e-045d562610356b4e"],"text":"A286718 \u2192 basis A013611 \u2192 D"},"headline":"A286718 \u2192 basis A013611 \u2192 D","chain":null}