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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"f64d2bdd8fad15e25611f9fb861361682585fb240c0394842fdde53912cf983d","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-629fc911ba71fa1d","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:          3;\n  2:         33,          36;\n  3:        366,         846,          486;\n  4:       4554,       16398,        19602,         7776;\n  5:      64269,      316998,       584577,       477576,        145800;\n  6:    1022607,     6434658,     16158447,     20238336,      12640860,       3149280;\n  7:   18176940,   139469364,    445068432,    756011556,     720869220,     365788872,      77157360;\n  8:  357656580,  3243472308,  12587060448,  27094386276,   34935123420,   26979444360,   11553763536,   2116316160;\n  9: 7725879495, 80939429172, 370514776986, 967927515372, 1578202881795, 1644507441648, 1069374004524, 396726139008, 64283103360;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(12*n^3-18*n^2+6*n)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(n - 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-7fd16a32c438d522","steps":["basis A013610","D"],"edges":["e-07b12f78e33af42b","e-1ba9a228a89bd875"],"text":"A059364 \u2192 basis A013610 \u2192 D"},"headline":"A059364 \u2192 basis A013610 \u2192 D","chain":null}