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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":"4ea639c30e4ef5b2d075d4055e2afd0482fd7d50b94cb3b303349a9d0c53cd3f","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-3255c4dcfe77aa0b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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:           2;\n  2:           6,           8;\n  3:          34,          64,          24;\n  4:         360,         776,         432,          64;\n  5:        6320,       14632,        9672,        2240,        160;\n  6:      167720,      405712,      295368,       84032,       9760,       384;\n  7:     6294040,    15682224,    12145752,     3896832,     571200,     37632,      896;\n  8:   317972800,   809287360,   654334272,   227230272,   38302080,   3252480,   132608,   2048;\n  9: 20831772000, 53875569600, 44959589760, 16514859072, 3057710880, 303336192, 16208640, 436224, 4608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(4*n^2-16*n+20)*x+4*x^2)*P(n-2,x) + (2*n^2-6*n+7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x + 4*x^2)*P[n - 2, x] + (2*n^2 - 6*n + 7 + 4*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x + 4*x^2)*v[i - 2] + (2*n^2 - 6*n + 7 + 4*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-54284d24a094e2d3","steps":["basis A013609","D"],"edges":["e-327e7f94e2e44f03","e-5532b8b59bc0ee7a"],"text":"A269944 \u2192 basis A013609 \u2192 D"},"headline":"A269944 \u2192 basis A013609 \u2192 D","chain":null}