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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":"457f5a3b4490ad2cbf9feb8e3af66790b2ba02673e0e15f6fc893d678d86cace","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-519925749fed9cdb","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*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:           1;\n  1:           5,            3;\n  2:          36,           42,            9;\n  3:         396,          600,          243,          27;\n  4:        5904,        10416,         5688,        1188,          81;\n  5:      111312,       217872,       142920,       41040,        5265,        243;\n  6:     2539584,      5370336,      4003056,     1406160,      251100,      21870,       729;\n  7:    68025024,    152870400,    125434512,    50824368,    11237940,    1374408,     86751,     2187;\n  8:  2092642560,   4942370304,   4374860544,  1975397760,   509138784,   77783328,   6940080,   332424,    6561;\n  9: 72706602240, 179039690496, 168625338624, 82978642944, 24045825888, 4315749984, 484254288, 32962464, 1240029, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-48*n+36+(24*n-36)*x+9*x^2)*P(n-2,x) + (8*n-8+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 48*n + 36 + (24*n - 36)*x + 9*x^2)*P[n - 2, x] + (8*n - 8 + 6*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] = 1; v[2] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 48*n + 36 + (24*n - 36)*x + 9*x^2)*v[i - 2] + (8*n - 8 + 6*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-4172472abe953cea","steps":["basis A013610","D+1"],"edges":["e-b01e725a7bd6431f","e-3442a6ed53fcb72a"],"text":"A290319 \u2192 basis A013610 \u2192 D+1"},"headline":"A290319 \u2192 basis A013610 \u2192 D+1","chain":null}