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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":"3d93c45bd2e042436c1ff510aa3b1d98f4657215225381e524704953f41e8c9e","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-3272d5bc697b9cb4","role":"sequence","name":"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:      2,      1;\n   3:      4,      5,       1;\n   4:      9,     20,       9,      1;\n   5:     25,     82,      59,     14,      1;\n   6:     90,    379,     370,    135,     20,     1;\n   7:    416,   2043,    2449,   1185,    265,    27,     1;\n   8:   2373,  12858,   17780,  10479,   3080,   469,    35,    1;\n   9:  16073,  93308,  143198,  97496,  34839,  6944,   770,   44,  1;\n  10: 125674, 769053, 1278468, 971690, 402066, 97419, 14112, 1194, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-11*n+15+(2*n-6)*x)*P(n-3,x) - (n^2-2*n-1+(2*n-4)*x+x^2)*P(n-2,x) + (2*n-3+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 4 + 5*x + x^2, P(4,x) = 9 + 20*x + 9*x^2 + x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[3, x_] := 4 + 5*x + x^2;\nP[4, x_] := 9 + 20*x + 9*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 11*n + 15 + (2*n - 6)*x)*P[n - 3, x] - (n^2 - 2*n - 1 + (2*n - 4)*x + x^2)*P[n - 2, x] + (2*n - 3 + 2*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] = 2 + x; v[3] = 4 + 5*x + x^2; v[4] = 9 + 20*x + 9*x^2 + x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 11*n + 15 + (2*n - 6)*x)*v[i - 3] - (n^2 - 2*n - 1 + (2*n - 4)*x + x^2)*v[i - 2] + (2*n - 3 + 2*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":["D+1"],"edges":["e-63225490f85a11ed"],"text":"A094816 \u2192 D+1"},"headline":"A094816 \u2192 D+1","chain":null}