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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":"56fe8701004ca54714bbd398bf6e0a3e5451332046ced68eda0360ab99ebda2a","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-b8883330fe0b24f8","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:         1;\n  1:         5,          2;\n  2:        29,         29,          6;\n  3:       193,        329,        170,         24;\n  4:      1458,       3604,       3155,       1114,        120;\n  5:     12366,      40564,      51227,      30532,       8244,        720;\n  6:    116568,     479996,     801277,     686537,     312184,      68508,       5040;\n  7:   1210392,    6023772,   12585813,   14214337,    9268704,    3420188,     633744,     40320;\n  8:  13736880,   80391816,  202548768,  285680896,  245032029,  129420692,   40282668,   6469776,   362880;\n  9: 169249680, 1140903576, 3374301288, 5729024536, 6126101769, 4249223862, 1890471428, 509661576, 72318240, 3628800;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(2*n^3+4*n^2+2*n)*x+(n^3+n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(2*n^2+2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (2*n^3 + 4*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (2*n^3 + 4*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (2*n^2 + 2*n)*x)*v[i - 1])/(n)); 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-2feb6dc01f424b0a","steps":["basis A007318","D+1"],"edges":["e-5281dccfa5f22cf2","e-838cc748a8c7f345"],"text":"A145324 \u2192 basis A007318 \u2192 D+1"},"headline":"A145324 \u2192 basis A007318 \u2192 D+1","chain":null}