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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":"7b474325b3f6a79036f306c2039feaa56021972cf0313957dec9359579fc65d7","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-1eee05e015d7347b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","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:      20,       33,         9;\n   4:      85,      231,       162,        27;\n   5:     413,     1518,      1719,       702,        81;\n   6:    2333,    10467,     15975,     10395,      2835,      243;\n   7:   15244,    78699,    147321,    128925,     55485,    10935,      729;\n   8:  113825,   652383,   1416933,   1525608,    881685,   272160,    40824,    2187;\n   9:  958817,  5962404,  14506686,  18219600,  12982599,  5375160,  1255338,  148716,   6561;\n  10: 9006373, 59829075, 159288390, 225461286, 188108487, 96176241, 30097494, 5524362, 531441, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(6*n-18)*x)*P(n-3,x) - (n^2-4+(6*n-6)*x+9*x^2)*P(n-2,x) + (2*n-1+6*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 5 + 3*x, P(3,x) = 20 + 33*x + 9*x^2, P(4,x) = 85 + 231*x + 162*x^2 + 27*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 3*x;\nP[3, x_] := 20 + 33*x + 9*x^2;\nP[4, x_] := 85 + 231*x + 162*x^2 + 27*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (6*n - 18)*x)*P[n - 3, x] - (n^2 - 4 + (6*n - 6)*x + 9*x^2)*P[n - 2, x] + (2*n - 1 + 6*x)*P[n - 1, x]];\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; v[3] = 20 + 33*x + 9*x^2; v[4] = 85 + 231*x + 162*x^2 + 27*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (6*n - 18)*x)*v[i - 3] - (n^2 - 4 + (6*n - 6)*x + 9*x^2)*v[i - 2] + (2*n - 1 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["basis A013610","D+1"],"edges":["e-792aa57b19fb3b04","e-dee615b2831d62d6"],"text":"A094816 \u2192 basis A013610 \u2192 D+1"},"headline":"A094816 \u2192 basis A013610 \u2192 D+1","chain":null}