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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":"5d0d1646cd5b71189c417f3e9b6afd3b892860e1e1e916d3c81bf2c8a06e4180","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-b0592fccca597966","role":"sequence","name":"basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 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:         15,          18;\n  3:        117,         216,          81;\n  4:       1254,        2862,        1782,        324;\n  5:      17142,       44730,       36045,      11340,       1215;\n  6:     285192,      818532,      778005,     325620,      61965,      4374;\n  7:    5593368,    17263260,    18465489,    9298800,    2398410,    306180,    15309;\n  8:  126373776,   413351928,   483925428,  278435556,   87635520,  15370236,  1408428,   52488;\n  9: 3232377360, 11092040856, 13958219376, 8896590612, 3235021335, 699743772, 89006526, 6141096, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(18*n-30)*x+9*x^2)*P(n-2,x) + (6*n-7+6*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[-(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*P[n - 2, x] + (6*n - 7 + 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] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*v[i - 2] + (6*n - 7 + 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-1a0f876fa1441737","steps":["basis A013610","D"],"edges":["e-6e846af4747cc041","e-03ca5382593ade50"],"text":"basis-map coefficients of A371080, shifted \u2192 basis A013610 \u2192 D"},"headline":"basis-map coefficients of A371080, shifted \u2192 basis A013610 \u2192 D","chain":null}