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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":"7e75ef7112c22391c27064f48b77f733fc5d833ff634316ab198016225382c55","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-831045384d9c95c9","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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:       5,        3;\n   3:      32,       42,         9;\n   4:     176,      468,       243,        27;\n   5:     896,     3648,      4032,      1188,        81;\n   6:    4352,    24000,     42480,     27000,      5265,       243;\n   7:   20480,   142848,    352512,    368928,    155520,     21870,      729;\n   8:   94208,   795648,   2532096,   3773952,   2667168,    809676,    86751,     2187;\n   9:  425984,  4227072,  16515072,  32237568,  32970240,  16982784,  3919104,   332424,    6561;\n  10: 1900544, 21676032, 100528128, 243689472, 330511104, 249422976, 98467488, 17950896, 1240029, 19683;\n","recurrence_sentence":"The row polynomials satisfy (6*n^3-35*n^2+66*n-40+(12*n^2-55*n+62)*x+(6*n-12)*x^2)*P(n,x) = (24*n^3-116*n^2+156*n-40+(54*n^3-231*n^2+203*n+86)*x+(108*n^2-417*n+402)*x^2+(54*n-108)*x^3)*P(n-1,x) - ((12*n^2-28*n+8)*x+(36*n^2-60*n-14)*x^2+(72*n-102)*x^3+36*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((24*n^3 - 116*n^2 + 156*n - 40 + (54*n^3 - 231*n^2 + 203*n + 86)*x + (108*n^2 - 417*n + 402)*x^2 + (54*n - 108)*x^3)*P[n - 1, x] - ((12*n^2 - 28*n + 8)*x + (36*n^2 - 60*n - 14)*x^2 + (72*n - 102)*x^3 + 36*x^4)*D[P[n - 1, x], x])/(6*n^3 - 35*n^2 + 66*n - 40 + (12*n^2 - 55*n + 62)*x + (6*n - 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = ((24*n^3 - 116*n^2 + 156*n - 40 + (54*n^3 - 231*n^2 + 203*n + 86)*x + (108*n^2 - 417*n + 402)*x^2 + (54*n - 108)*x^3)*v[i - 1] - ((12*n^2 - 28*n + 8)*x + (36*n^2 - 60*n - 14)*x^2 + (72*n - 102)*x^3 + 36*x^4)*deriv(v[i - 1], x))/(6*n^3 - 35*n^2 + 66*n - 40 + (12*n^2 - 55*n + 62)*x + (6*n - 12)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013610","D+1"],"edges":["e-3b5bae8e6d048f8c","e-65ae5d96832185d0"],"text":"A086645 \u2192 basis A013610 \u2192 D+1"},"headline":"A086645 \u2192 basis A013610 \u2192 D+1","chain":null}