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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":"310f3a77d56b35efd0ea33863355cda9648ed5d5712654c7c49afed4a5a771b4","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-6ace63f6e12c5046","role":"sequence","name":"Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","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:         1;\n  2:         9,          2;\n  3:        83,         36,         3;\n  4:       870,        550,        90,         4;\n  5:     10474,       8550,      2055,       180,        5;\n  6:    143892,     141848,     44415,      5740,      315,       6;\n  7:   2233356,    2548392,    972867,    164640,    13370,     504,     7;\n  8:  38769840,   49741144,  22196916,   4605636,   491400,   27468,   756,    8;\n  9: 745785936, 1053582120, 533835600, 130239900, 17099565, 1258740, 51450, 1080, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^4-32*n^3+92*n^2-112*n+48)*P(n-4,x) + (12*n^3-54*n^2+78*n-36+(4*n^2-12*n+8)*x)*P(n-3,x) - (13*n^2-26*n+13+(6*n-6)*x+x^2)*P(n-2,x) + (6*n-3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 9 + 2*x, P(3,x) = 83 + 36*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 9 + 2*x;\nP[3, x_] := 83 + 36*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(4*n^4 - 32*n^3 + 92*n^2 - 112*n + 48)*P[n - 4, x] + (12*n^3 - 54*n^2 + 78*n - 36 + (4*n^2 - 12*n + 8)*x)*P[n - 3, x] - (13*n^2 - 26*n + 13 + (6*n - 6)*x + x^2)*P[n - 2, x] + (6*n - 3 + 2*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] = 1; v[3] = 9 + 2*x; v[4] = 83 + 36*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 32*n^3 + 92*n^2 - 112*n + 48)*v[i - 4] + (12*n^3 - 54*n^2 + 78*n - 36 + (4*n^2 - 12*n + 8)*x)*v[i - 3] - (13*n^2 - 26*n + 13 + (6*n - 6)*x + x^2)*v[i - 2] + (6*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6ace63f6e12c5046","steps":["D"],"edges":["e-222e2f80fe891e92"],"text":"A079638 \u2192 D"},"headline":"A079638 \u2192 D","chain":null}